| L(s) = 1 | + 5.26·2-s + 19.6·4-s + 5·5-s − 10.3·7-s + 61.4·8-s + 26.3·10-s − 11·11-s + 63.9·13-s − 54.3·14-s + 165.·16-s − 17.1·17-s + 90.2·19-s + 98.4·20-s − 57.8·22-s + 212.·23-s + 25·25-s + 336.·26-s − 203.·28-s − 57.5·29-s − 141.·31-s + 381.·32-s − 90.2·34-s − 51.6·35-s − 257.·37-s + 474.·38-s + 307.·40-s + 225.·41-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.46·4-s + 0.447·5-s − 0.557·7-s + 2.71·8-s + 0.831·10-s − 0.301·11-s + 1.36·13-s − 1.03·14-s + 2.59·16-s − 0.244·17-s + 1.08·19-s + 1.10·20-s − 0.560·22-s + 1.92·23-s + 0.200·25-s + 2.53·26-s − 1.37·28-s − 0.368·29-s − 0.820·31-s + 2.10·32-s − 0.455·34-s − 0.249·35-s − 1.14·37-s + 2.02·38-s + 1.21·40-s + 0.860·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.914462870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.914462870\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 5.26T + 8T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 13 | \( 1 - 63.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 212.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 225.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 404.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 259.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 92.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 242.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 706.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 440.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 197.T + 9.12e5T^{2} \) |
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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
−11.04034650429557067793682428924, −9.908040291813923333466396194314, −8.705866672315565271473547443776, −7.25618065771403870770627620592, −6.55461965530677026412684506762, −5.63960419299917378855309782888, −4.94399065267532515937127494808, −3.60843608193586126963928075138, −2.97787144440632533449869752915, −1.51168657565424928911376892449,
1.51168657565424928911376892449, 2.97787144440632533449869752915, 3.60843608193586126963928075138, 4.94399065267532515937127494808, 5.63960419299917378855309782888, 6.55461965530677026412684506762, 7.25618065771403870770627620592, 8.705866672315565271473547443776, 9.908040291813923333466396194314, 11.04034650429557067793682428924
