| L(s) = 1 | + 3-s + 0.388·5-s − 7-s + 9-s − 6.41·11-s + 3.88·13-s + 0.388·15-s − 4.98·17-s + 19-s − 21-s − 3.44·23-s − 4.84·25-s + 27-s + 0.169·29-s + 8.62·31-s − 6.41·33-s − 0.388·35-s + 7.37·37-s + 3.88·39-s − 8.77·41-s + 9.11·43-s + 0.388·45-s + 4.80·47-s + 49-s − 4.98·51-s + 8.42·53-s − 2.49·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.173·5-s − 0.377·7-s + 0.333·9-s − 1.93·11-s + 1.07·13-s + 0.100·15-s − 1.20·17-s + 0.229·19-s − 0.218·21-s − 0.717·23-s − 0.969·25-s + 0.192·27-s + 0.0315·29-s + 1.54·31-s − 1.11·33-s − 0.0656·35-s + 1.21·37-s + 0.622·39-s − 1.37·41-s + 1.38·43-s + 0.0578·45-s + 0.701·47-s + 0.142·49-s − 0.698·51-s + 1.15·53-s − 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979830259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979830259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 0.388T + 5T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 13 | \( 1 - 3.88T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 - 0.169T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 - 9.11T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 - 5.82T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 1.01T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.119080785289954541097164005074, −7.49735441137064854369387301996, −6.56902365448385343461061925181, −5.96462206349699154438969615298, −5.18944411953205440518802428040, −4.31315717351898651148083408160, −3.58449410218259649182804459177, −2.58216217199880873021572467585, −2.17517362970767110768414518571, −0.68178474640402712422350320288,
0.68178474640402712422350320288, 2.17517362970767110768414518571, 2.58216217199880873021572467585, 3.58449410218259649182804459177, 4.31315717351898651148083408160, 5.18944411953205440518802428040, 5.96462206349699154438969615298, 6.56902365448385343461061925181, 7.49735441137064854369387301996, 8.119080785289954541097164005074
