| L(s) = 1 | − 32·2-s − 90·3-s − 7.48e3·5-s + 2.88e3·6-s + 3.27e4·8-s + 1.77e5·9-s + 2.39e5·10-s + 2.94e5·11-s + 4.21e5·13-s + 6.73e5·15-s − 1.04e6·16-s − 6.96e6·17-s − 5.66e6·18-s − 9.34e6·19-s − 9.42e6·22-s − 5.11e7·23-s − 2.94e6·24-s + 4.88e7·25-s − 1.34e7·26-s − 4.71e7·27-s + 3.32e8·29-s − 2.15e7·30-s + 1.19e8·31-s − 2.65e7·33-s + 2.22e8·34-s + 2.75e8·37-s + 2.99e8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.213·3-s − 1.07·5-s + 0.151·6-s + 0.353·8-s + 9-s + 0.756·10-s + 0.551·11-s + 0.314·13-s + 0.228·15-s − 1/4·16-s − 1.18·17-s − 0.707·18-s − 0.865·19-s − 0.389·22-s − 1.65·23-s − 0.0756·24-s + 25-s − 0.222·26-s − 0.631·27-s + 3.00·29-s − 0.161·30-s + 0.746·31-s − 0.117·33-s + 0.841·34-s + 0.653·37-s + 0.612·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.01805761516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01805761516\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{5} T + p^{10} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 10 p^{2} T - 2087 p^{4} T^{2} + 10 p^{13} T^{3} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 1496 p T + 284891 p^{2} T^{2} + 1496 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 26776 p T - 1640993515 p^{2} T^{2} - 26776 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 210588 T + p^{11} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6962906 T + 14210163657203 T^{2} + 6962906 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 9346390 T - 29135252866119 T^{2} + 9346390 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 51172000 T + 1665763826086073 T^{2} + 51172000 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 166196354 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 119000988 T - 11247241751428687 T^{2} - 119000988 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 275545510 T - 101992293698300313 T^{2} - 275545510 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 197988378 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 809489728 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2600196204 T + 4288861084211997313 T^{2} + 2600196204 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 733631454 T - 8730820819074037481 T^{2} + 733631454 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4657126942 T - 8467057090835571295 T^{2} + 4657126942 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5135837424 T - 17137091565676882885 T^{2} + 5135837424 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8810564836 T - 44504080175608310187 T^{2} + 8810564836 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3849006656 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18686748254 T + 35467874740012340139 T^{2} + 18686748254 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 29850061992 T + \)\(14\!\cdots\!85\)\( T^{2} - 29850061992 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 5875980446 T + p^{11} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 83056539450 T + \)\(41\!\cdots\!11\)\( T^{2} - 83056539450 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 149400800374 T + p^{11} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.97938890791071754544440939570, −11.39840785972421025161444158111, −10.91163331550182870789713134863, −10.18298987290050879807783075052, −10.05659999146654199736166124069, −9.248974103556885047847269033125, −8.589917684813325771746275149225, −8.201207121421591346001941863110, −7.84025096288656968572792627874, −6.96480554776434516235328226452, −6.49653754466579629738501288912, −6.18241676488075675078142736135, −4.76242175191814446977851353844, −4.49894594858362981435360232866, −4.08818997385338072702485023428, −3.23753929933461550534341075905, −2.38678160072934322402287371184, −1.50038070324109622070596985456, −1.07327511330187070305006411400, −0.04188853926333191843741893889,
0.04188853926333191843741893889, 1.07327511330187070305006411400, 1.50038070324109622070596985456, 2.38678160072934322402287371184, 3.23753929933461550534341075905, 4.08818997385338072702485023428, 4.49894594858362981435360232866, 4.76242175191814446977851353844, 6.18241676488075675078142736135, 6.49653754466579629738501288912, 6.96480554776434516235328226452, 7.84025096288656968572792627874, 8.201207121421591346001941863110, 8.589917684813325771746275149225, 9.248974103556885047847269033125, 10.05659999146654199736166124069, 10.18298987290050879807783075052, 10.91163331550182870789713134863, 11.39840785972421025161444158111, 11.97938890791071754544440939570
