| L(s) = 1 | + 2·2-s − 5·3-s − 10·6-s + 28·7-s − 8·8-s + 27·9-s + 57·11-s + 140·13-s + 56·14-s − 16·16-s + 51·17-s + 54·18-s − 5·19-s − 140·21-s + 114·22-s + 69·23-s + 40·24-s + 280·26-s − 280·27-s + 228·29-s − 23·31-s − 285·33-s + 102·34-s − 253·37-s − 10·38-s − 700·39-s − 84·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.962·3-s − 0.680·6-s + 1.51·7-s − 0.353·8-s + 9-s + 1.56·11-s + 2.98·13-s + 1.06·14-s − 1/4·16-s + 0.727·17-s + 0.707·18-s − 0.0603·19-s − 1.45·21-s + 1.10·22-s + 0.625·23-s + 0.340·24-s + 2.11·26-s − 1.99·27-s + 1.45·29-s − 0.133·31-s − 1.50·33-s + 0.514·34-s − 1.12·37-s − 0.0426·38-s − 2.87·39-s − 0.319·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.285487342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.285487342\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 57 T + 1918 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 p T - 8 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T - 6834 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 p T - 14 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 23 T - 29262 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 253 T + 13356 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 201 T - 63422 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 393 T + 5572 T^{2} + 393 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 219 T - 157418 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 709 T + 275700 T^{2} - 709 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 419 T - 125202 T^{2} - 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 313 T - 291048 T^{2} + 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 461 T - 280518 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 588 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1017 T + 329320 T^{2} - 1017 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1834 T + p^{3} T^{2} )^{2} \) |
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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
−11.21307607014289187109037072916, −11.10120509860722805764954056191, −10.60501630831194219221665983919, −10.16374787402305466313065997166, −9.238108530593104452575772928186, −9.050606310313651037760441213752, −8.418548346751709060095335691810, −8.117134297795076286777946332493, −7.38534328075704442680927363536, −6.78553316966676088688357338147, −6.16125339040493935205267338272, −6.07426615345231567469040801933, −5.35233417248031440227197322745, −4.86992004788935374197038202736, −4.26564316463339545885994796397, −3.72126794100720122011573819414, −3.47558361068407025944971424801, −1.95825256698028243988393907467, −1.16699332580538066909847275053, −1.03866875286324919153150668894,
1.03866875286324919153150668894, 1.16699332580538066909847275053, 1.95825256698028243988393907467, 3.47558361068407025944971424801, 3.72126794100720122011573819414, 4.26564316463339545885994796397, 4.86992004788935374197038202736, 5.35233417248031440227197322745, 6.07426615345231567469040801933, 6.16125339040493935205267338272, 6.78553316966676088688357338147, 7.38534328075704442680927363536, 8.117134297795076286777946332493, 8.418548346751709060095335691810, 9.050606310313651037760441213752, 9.238108530593104452575772928186, 10.16374787402305466313065997166, 10.60501630831194219221665983919, 11.10120509860722805764954056191, 11.21307607014289187109037072916
