| L(s) = 1 | + 1.54·2-s − 1.51·3-s + 0.390·4-s − 2.33·6-s + 4.16·7-s − 2.48·8-s − 0.709·9-s − 11-s − 0.591·12-s − 1.82·13-s + 6.43·14-s − 4.62·16-s − 4.80·17-s − 1.09·18-s + 19-s − 6.29·21-s − 1.54·22-s + 5.53·23-s + 3.76·24-s − 2.82·26-s + 5.61·27-s + 1.62·28-s + 3.98·29-s − 8.51·31-s − 2.18·32-s + 1.51·33-s − 7.42·34-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 0.873·3-s + 0.195·4-s − 0.955·6-s + 1.57·7-s − 0.879·8-s − 0.236·9-s − 0.301·11-s − 0.170·12-s − 0.507·13-s + 1.71·14-s − 1.15·16-s − 1.16·17-s − 0.258·18-s + 0.229·19-s − 1.37·21-s − 0.329·22-s + 1.15·23-s + 0.768·24-s − 0.554·26-s + 1.08·27-s + 0.307·28-s + 0.740·29-s − 1.52·31-s − 0.385·32-s + 0.263·33-s − 1.27·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.101971734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101971734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 - 7.11T + 37T^{2} \) |
| 41 | \( 1 + 3.85T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.75T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 - 6.03T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 + 0.508T + 83T^{2} \) |
| 89 | \( 1 - 6.86T + 89T^{2} \) |
| 97 | \( 1 - 3.72T + 97T^{2} \) |
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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
−8.168519176255401261374103806777, −7.30062587542652261655777027572, −6.53188030316966516419859582243, −5.74121577659004146190715647079, −5.08519057331691638717858149216, −4.81118916850653398393487530171, −4.07748362673443744149618268464, −2.91719653730394109141722021697, −2.09305191637250515281931168864, −0.68512951077828674650859365684,
0.68512951077828674650859365684, 2.09305191637250515281931168864, 2.91719653730394109141722021697, 4.07748362673443744149618268464, 4.81118916850653398393487530171, 5.08519057331691638717858149216, 5.74121577659004146190715647079, 6.53188030316966516419859582243, 7.30062587542652261655777027572, 8.168519176255401261374103806777
