| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s − 3·11-s + 12-s − 4·13-s + 4·14-s + 16-s + 3·17-s − 18-s − 4·19-s − 4·21-s + 3·22-s − 24-s + 4·26-s + 27-s − 4·28-s − 4·31-s − 32-s − 3·33-s − 3·34-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s − 0.872·21-s + 0.639·22-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.755·28-s − 0.718·31-s − 0.176·32-s − 0.522·33-s − 0.514·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4775630283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4775630283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.08762853170164, −13.23285428742647, −13.09392553088713, −12.48694243633612, −12.22727053137392, −11.47525034012899, −10.67748616424363, −10.46580390199173, −9.762405649564950, −9.591080936735980, −9.093116546155042, −8.462319228262863, −7.732157628296994, −7.635126071667921, −6.878860172509083, −6.428026640552064, −5.840219528925147, −5.212709704763796, −4.485286485116356, −3.715262833978267, −3.207250309009053, −2.460044561261555, −2.351755701884437, −1.197245041733002, −0.2500498281034541,
0.2500498281034541, 1.197245041733002, 2.351755701884437, 2.460044561261555, 3.207250309009053, 3.715262833978267, 4.485286485116356, 5.212709704763796, 5.840219528925147, 6.428026640552064, 6.878860172509083, 7.635126071667921, 7.732157628296994, 8.462319228262863, 9.093116546155042, 9.591080936735980, 9.762405649564950, 10.46580390199173, 10.67748616424363, 11.47525034012899, 12.22727053137392, 12.48694243633612, 13.09392553088713, 13.23285428742647, 14.08762853170164
