| L(s) = 1 | + 4.09e3·2-s + 1.75e6·3-s + 1.67e7·4-s − 1.37e8·5-s + 7.20e9·6-s − 3.04e10·7-s + 6.87e10·8-s + 2.24e12·9-s − 5.61e11·10-s + 2.58e12·11-s + 2.94e13·12-s − 9.57e13·13-s − 1.24e14·14-s − 2.40e14·15-s + 2.81e14·16-s − 1.64e15·17-s + 9.18e15·18-s + 4.95e15·19-s − 2.29e15·20-s − 5.34e16·21-s + 1.05e16·22-s − 1.07e16·23-s + 1.20e17·24-s − 2.79e17·25-s − 3.92e17·26-s + 2.45e18·27-s − 5.10e17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s − 0.251·5-s + 1.35·6-s − 0.830·7-s + 0.353·8-s + 2.64·9-s − 0.177·10-s + 0.248·11-s + 0.954·12-s − 1.13·13-s − 0.587·14-s − 0.479·15-s + 0.250·16-s − 0.685·17-s + 1.87·18-s + 0.513·19-s − 0.125·20-s − 1.58·21-s + 0.175·22-s − 0.102·23-s + 0.675·24-s − 0.936·25-s − 0.805·26-s + 3.14·27-s − 0.415·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.873599089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.873599089\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| good | 3 | \( 1 - 1.75e6T + 8.47e11T^{2} \) |
| 5 | \( 1 + 1.37e8T + 2.98e17T^{2} \) |
| 7 | \( 1 + 3.04e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 2.58e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 9.57e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.64e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 4.95e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.07e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.36e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.41e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.01e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.58e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.83e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 1.40e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 1.99e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 4.16e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.42e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 8.67e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 5.13e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.49e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.91e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.64e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 8.74e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.00e25T + 4.66e49T^{2} \) |
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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
−21.97766147046738826075335824364, −20.24180583354712465316532818185, −19.29205137885697460875049351785, −15.68178921653690092362764057562, −14.31572441593052936437737236132, −12.85583118836617094852809924520, −9.477013863951213799278245625209, −7.38952923442123633934034670144, −3.86947813471126057571003514132, −2.40178545892333212608839019732,
2.40178545892333212608839019732, 3.86947813471126057571003514132, 7.38952923442123633934034670144, 9.477013863951213799278245625209, 12.85583118836617094852809924520, 14.31572441593052936437737236132, 15.68178921653690092362764057562, 19.29205137885697460875049351785, 20.24180583354712465316532818185, 21.97766147046738826075335824364
