| L(s) = 1 | + 9.82·3-s + 0.312·7-s + 69.6·9-s − 42.5·11-s − 42.4·13-s − 41.1·17-s − 90.1·19-s + 3.07·21-s − 23·23-s + 418.·27-s − 277.·29-s + 197.·31-s − 418.·33-s − 11.8·37-s − 417.·39-s + 401.·41-s + 288.·43-s + 29.1·47-s − 342.·49-s − 404.·51-s − 459.·53-s − 886.·57-s − 340.·59-s − 683.·61-s + 21.7·63-s − 337.·67-s − 226.·69-s + ⋯ |
| L(s) = 1 | + 1.89·3-s + 0.0168·7-s + 2.57·9-s − 1.16·11-s − 0.905·13-s − 0.587·17-s − 1.08·19-s + 0.0319·21-s − 0.208·23-s + 2.98·27-s − 1.77·29-s + 1.14·31-s − 2.20·33-s − 0.0528·37-s − 1.71·39-s + 1.52·41-s + 1.02·43-s + 0.0903·47-s − 0.999·49-s − 1.11·51-s − 1.19·53-s − 2.06·57-s − 0.751·59-s − 1.43·61-s + 0.0435·63-s − 0.615·67-s − 0.394·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 9.82T + 27T^{2} \) |
| 7 | \( 1 - 0.312T + 343T^{2} \) |
| 11 | \( 1 + 42.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 401.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 288.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 29.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 459.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 337.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 610.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 776.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 113.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−8.102686528573972303050196451583, −7.78746416320556233181900490278, −7.09374099368623440921146276873, −6.00959748117769717872996540709, −4.70037938103469521445448120153, −4.18457882049177425344350699213, −3.02772029965330995271901925787, −2.47865608056401810769896843883, −1.70741734863882025214566352551, 0,
1.70741734863882025214566352551, 2.47865608056401810769896843883, 3.02772029965330995271901925787, 4.18457882049177425344350699213, 4.70037938103469521445448120153, 6.00959748117769717872996540709, 7.09374099368623440921146276873, 7.78746416320556233181900490278, 8.102686528573972303050196451583
