| L(s) = 1 | + 4.09e3·2-s − 7.15e5·3-s + 1.67e7·4-s − 2.44e8·5-s − 2.93e9·6-s − 6.40e10·7-s + 6.87e10·8-s − 3.35e11·9-s − 1.00e12·10-s + 1.40e13·11-s − 1.20e13·12-s + 1.57e13·13-s − 2.62e14·14-s + 1.74e14·15-s + 2.81e14·16-s − 1.00e15·17-s − 1.37e15·18-s − 8.66e15·19-s − 4.09e15·20-s + 4.58e16·21-s + 5.75e16·22-s + 1.43e17·23-s − 4.91e16·24-s + 5.96e16·25-s + 6.45e16·26-s + 8.46e17·27-s − 1.07e18·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.777·3-s + 0.5·4-s − 0.447·5-s − 0.549·6-s − 1.74·7-s + 0.353·8-s − 0.395·9-s − 0.316·10-s + 1.35·11-s − 0.388·12-s + 0.187·13-s − 1.23·14-s + 0.347·15-s + 0.250·16-s − 0.419·17-s − 0.279·18-s − 0.898·19-s − 0.223·20-s + 1.35·21-s + 0.955·22-s + 1.36·23-s − 0.274·24-s + 0.199·25-s + 0.132·26-s + 1.08·27-s − 0.874·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.539905076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539905076\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 5 | \( 1 + 2.44e8T \) |
| good | 3 | \( 1 + 7.15e5T + 8.47e11T^{2} \) |
| 7 | \( 1 + 6.40e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.40e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.57e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.00e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 8.66e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.43e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 3.60e17T + 3.63e36T^{2} \) |
| 31 | \( 1 - 8.72e17T + 1.92e37T^{2} \) |
| 37 | \( 1 - 7.37e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 6.57e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 4.50e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 5.55e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.75e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.95e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.95e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 4.26e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.93e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.11e22T + 3.82e46T^{2} \) |
| 79 | \( 1 + 4.99e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.79e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.07e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 3.44e24T + 4.66e49T^{2} \) |
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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
−14.87198458028148213054408099432, −13.17722026568140347982586287076, −12.10701551749495798808679016525, −10.92490040001171314166845916862, −9.150923791724951314718736346031, −6.77611697309666041293193480564, −6.03026217565657804221099389387, −4.20830786797002826222917943676, −2.94693380386204346244762430629, −0.67250797503421422148198550273,
0.67250797503421422148198550273, 2.94693380386204346244762430629, 4.20830786797002826222917943676, 6.03026217565657804221099389387, 6.77611697309666041293193480564, 9.150923791724951314718736346031, 10.92490040001171314166845916862, 12.10701551749495798808679016525, 13.17722026568140347982586287076, 14.87198458028148213054408099432
