| L(s) = 1 | + 3·3-s − 12.8·7-s + 9·9-s − 49.7·11-s + 52.6·13-s − 84.6·17-s + 26.6·19-s − 38.6·21-s + 136.·23-s + 27·27-s + 6·29-s − 47.1·31-s − 149.·33-s − 344.·37-s + 157.·39-s − 43.2·41-s + 252·43-s + 306.·47-s − 177.·49-s − 253.·51-s + 455.·53-s + 79.8·57-s − 708.·59-s − 652.·61-s − 115.·63-s + 704.·67-s + 409.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.694·7-s + 0.333·9-s − 1.36·11-s + 1.12·13-s − 1.20·17-s + 0.321·19-s − 0.401·21-s + 1.23·23-s + 0.192·27-s + 0.0384·29-s − 0.273·31-s − 0.787·33-s − 1.52·37-s + 0.647·39-s − 0.164·41-s + 0.893·43-s + 0.949·47-s − 0.517·49-s − 0.696·51-s + 1.17·53-s + 0.185·57-s − 1.56·59-s − 1.36·61-s − 0.231·63-s + 1.28·67-s + 0.713·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.056992483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056992483\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 12.8T + 343T^{2} \) |
| 11 | \( 1 + 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 43.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252T + 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 455.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 708.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 652.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 704.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 531.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 57.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 429.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 227.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 152.T + 9.12e5T^{2} \) |
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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
−8.835504663177539440513298629637, −7.85847149432788064754402904429, −7.13546071795115031793995715926, −6.37820455433755978716958522793, −5.46810911643053018248133776243, −4.60216540002311463252748140077, −3.54518681522213849097116523563, −2.89717962968805841201343684729, −1.93941681102257010572599621685, −0.59955939882341680212711789625,
0.59955939882341680212711789625, 1.93941681102257010572599621685, 2.89717962968805841201343684729, 3.54518681522213849097116523563, 4.60216540002311463252748140077, 5.46810911643053018248133776243, 6.37820455433755978716958522793, 7.13546071795115031793995715926, 7.85847149432788064754402904429, 8.835504663177539440513298629637
