| L(s) = 1 | + 2-s − 3-s + 4-s − 1.61·5-s − 6-s + 8-s + 9-s − 1.61·10-s − 11-s − 12-s + 1.61·15-s + 16-s − 7.34·17-s + 18-s + 1.17·19-s − 1.61·20-s − 22-s + 3.55·23-s − 24-s − 2.39·25-s − 27-s + 10.3·29-s + 1.61·30-s + 6.34·31-s + 32-s + 33-s − 7.34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.721·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.510·10-s − 0.301·11-s − 0.288·12-s + 0.416·15-s + 0.250·16-s − 1.78·17-s + 0.235·18-s + 0.268·19-s − 0.360·20-s − 0.213·22-s + 0.741·23-s − 0.204·24-s − 0.479·25-s − 0.192·27-s + 1.93·29-s + 0.294·30-s + 1.13·31-s + 0.176·32-s + 0.174·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.855105970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855105970\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 1.61T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4.94T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 4.39T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 - 5.05T + 83T^{2} \) |
| 89 | \( 1 + 0.773T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.473650396790543273812909141406, −7.84553452701584066735953834708, −6.83537775653755787055218829842, −6.53097533673180727057813588425, −5.51603538273905658558150676924, −4.63538443336979284942494968149, −4.27568216908515599109311923907, −3.14767287423707492032674949862, −2.24077992026809123306857026515, −0.74561746532090103070418096808,
0.74561746532090103070418096808, 2.24077992026809123306857026515, 3.14767287423707492032674949862, 4.27568216908515599109311923907, 4.63538443336979284942494968149, 5.51603538273905658558150676924, 6.53097533673180727057813588425, 6.83537775653755787055218829842, 7.84553452701584066735953834708, 8.473650396790543273812909141406
