| L(s) = 1 | + 3·3-s + 5·5-s + 11.0·7-s + 9·9-s − 35.6·11-s + 13·13-s + 15·15-s + 9.34·17-s − 77.5·19-s + 33.1·21-s − 204.·23-s + 25·25-s + 27·27-s + 176.·29-s − 216.·31-s − 106.·33-s + 55.3·35-s − 190.·37-s + 39·39-s − 315.·41-s + 33.6·43-s + 45·45-s − 203.·47-s − 220.·49-s + 28.0·51-s + 487.·53-s − 178.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.597·7-s + 0.333·9-s − 0.977·11-s + 0.277·13-s + 0.258·15-s + 0.133·17-s − 0.936·19-s + 0.344·21-s − 1.85·23-s + 0.200·25-s + 0.192·27-s + 1.12·29-s − 1.25·31-s − 0.564·33-s + 0.267·35-s − 0.847·37-s + 0.160·39-s − 1.20·41-s + 0.119·43-s + 0.149·45-s − 0.630·47-s − 0.643·49-s + 0.0770·51-s + 1.26·53-s − 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 - 11.0T + 343T^{2} \) |
| 11 | \( 1 + 35.6T + 1.33e3T^{2} \) |
| 17 | \( 1 - 9.34T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 216.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 203.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 487.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 176.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 234.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 608.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 454.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 427.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 821.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 549.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 213.T + 9.12e5T^{2} \) |
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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
−8.443186693911619093209876693833, −8.141156448121415244374438233919, −7.16923955893145807212499751255, −6.20804622333751156887636899405, −5.34110595224487665283659628317, −4.46505012928887216778930210016, −3.46188728105427891209651968631, −2.33293124724066647675523968363, −1.63047349787470971457486970262, 0,
1.63047349787470971457486970262, 2.33293124724066647675523968363, 3.46188728105427891209651968631, 4.46505012928887216778930210016, 5.34110595224487665283659628317, 6.20804622333751156887636899405, 7.16923955893145807212499751255, 8.141156448121415244374438233919, 8.443186693911619093209876693833
