L(s) = 1 | + 1.61·2-s + 0.618·4-s + 2.23·5-s − 7-s − 2.23·8-s + 3.61·10-s − 0.236·13-s − 1.61·14-s − 4.85·16-s + 2·17-s + 1.47·19-s + 1.38·20-s + 2·23-s − 0.381·26-s − 0.618·28-s + 5·29-s + 10.4·31-s − 3.38·32-s + 3.23·34-s − 2.23·35-s − 1.47·37-s + 2.38·38-s − 5.00·40-s + 2.47·41-s + 8.47·43-s + 3.23·46-s − 9.47·47-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.309·4-s + 0.999·5-s − 0.377·7-s − 0.790·8-s + 1.14·10-s − 0.0654·13-s − 0.432·14-s − 1.21·16-s + 0.485·17-s + 0.337·19-s + 0.309·20-s + 0.417·23-s − 0.0749·26-s − 0.116·28-s + 0.928·29-s + 1.88·31-s − 0.597·32-s + 0.554·34-s − 0.377·35-s − 0.242·37-s + 0.386·38-s − 0.790·40-s + 0.386·41-s + 1.29·43-s + 0.477·46-s − 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.966334855\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.966334855\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 0.472T + 79T^{2} \) |
| 83 | \( 1 - 7.52T + 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81831357650664007853897358258, −6.82764814858202645336872175834, −6.16340696168307906343622535329, −5.88323085076054928860637490054, −4.89890866966525158300282345702, −4.58978580035627707297643587465, −3.44453051566755071249069025030, −2.93850217898377106423188940653, −2.08770197072324305690088884815, −0.840012123146977486997558457768,
0.840012123146977486997558457768, 2.08770197072324305690088884815, 2.93850217898377106423188940653, 3.44453051566755071249069025030, 4.58978580035627707297643587465, 4.89890866966525158300282345702, 5.88323085076054928860637490054, 6.16340696168307906343622535329, 6.82764814858202645336872175834, 7.81831357650664007853897358258