| L(s) = 1 | − 8.54·3-s + 12.7·5-s + 23.4·7-s + 46.0·9-s − 11.4·13-s − 108.·15-s + 65.5·17-s − 7.25·19-s − 200.·21-s + 104.·23-s + 37.2·25-s − 162.·27-s + 127.·29-s + 288.·31-s + 298.·35-s − 85.4·37-s + 97.9·39-s + 135.·41-s + 353.·43-s + 586.·45-s + 134.·47-s + 207.·49-s − 559.·51-s + 501.·53-s + 61.9·57-s − 651.·59-s − 365.·61-s + ⋯ |
| L(s) = 1 | − 1.64·3-s + 1.13·5-s + 1.26·7-s + 1.70·9-s − 0.244·13-s − 1.87·15-s + 0.934·17-s − 0.0875·19-s − 2.08·21-s + 0.943·23-s + 0.297·25-s − 1.15·27-s + 0.815·29-s + 1.67·31-s + 1.44·35-s − 0.379·37-s + 0.402·39-s + 0.515·41-s + 1.25·43-s + 1.94·45-s + 0.417·47-s + 0.604·49-s − 1.53·51-s + 1.29·53-s + 0.143·57-s − 1.43·59-s − 0.767·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.153122645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153122645\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 8.54T + 27T^{2} \) |
| 5 | \( 1 - 12.7T + 125T^{2} \) |
| 7 | \( 1 - 23.4T + 343T^{2} \) |
| 13 | \( 1 + 11.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.25T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 501.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 651.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 365.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 294.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 408.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 260.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.941999499081796721037679532030, −7.918359138797018218080862476053, −7.08485782067964243953190288640, −6.19152442675581220367093914162, −5.64362300561275791647401935619, −4.97743521612482238104904821186, −4.38151959591655968317545807321, −2.67272431871954971370344091852, −1.47611102496785195145024188022, −0.835491562791308518257955501609,
0.835491562791308518257955501609, 1.47611102496785195145024188022, 2.67272431871954971370344091852, 4.38151959591655968317545807321, 4.97743521612482238104904821186, 5.64362300561275791647401935619, 6.19152442675581220367093914162, 7.08485782067964243953190288640, 7.918359138797018218080862476053, 8.941999499081796721037679532030
