L(s) = 1 | + 1.23·2-s − 0.477·4-s − 5-s − 2.41·7-s − 3.05·8-s − 1.23·10-s + 11-s + 5.99·13-s − 2.97·14-s − 2.81·16-s + 17-s − 3.44·19-s + 0.477·20-s + 1.23·22-s − 8.43·23-s + 25-s + 7.39·26-s + 1.15·28-s − 3.64·29-s + 8.98·31-s + 2.63·32-s + 1.23·34-s + 2.41·35-s + 4.91·37-s − 4.24·38-s + 3.05·40-s − 11.0·41-s + ⋯ |
L(s) = 1 | + 0.872·2-s − 0.238·4-s − 0.447·5-s − 0.912·7-s − 1.08·8-s − 0.390·10-s + 0.301·11-s + 1.66·13-s − 0.796·14-s − 0.704·16-s + 0.242·17-s − 0.789·19-s + 0.106·20-s + 0.263·22-s − 1.75·23-s + 0.200·25-s + 1.44·26-s + 0.217·28-s − 0.676·29-s + 1.61·31-s + 0.466·32-s + 0.211·34-s + 0.408·35-s + 0.807·37-s − 0.688·38-s + 0.483·40-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.628846674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.628846674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 13 | \( 1 - 5.99T + 13T^{2} \) |
| 19 | \( 1 + 3.44T + 19T^{2} \) |
| 23 | \( 1 + 8.43T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 - 8.98T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 + 4.63T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 - 9.30T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.984479186317769289476176970911, −6.64645908136662592617232837978, −6.37025228145976994603950014395, −5.82018421051110817529390254644, −4.87941221973578125664054152871, −4.10839852622914524406618618747, −3.61404817054045702547350281106, −3.11405031682352018596964604257, −1.88159339780975900416191821208, −0.53707860052745333487296778189,
0.53707860052745333487296778189, 1.88159339780975900416191821208, 3.11405031682352018596964604257, 3.61404817054045702547350281106, 4.10839852622914524406618618747, 4.87941221973578125664054152871, 5.82018421051110817529390254644, 6.37025228145976994603950014395, 6.64645908136662592617232837978, 7.984479186317769289476176970911