| L(s) = 1 | − 31.1·2-s − 183.·3-s + 459.·4-s − 1.66e3·5-s + 5.72e3·6-s + 2.40e3·7-s + 1.65e3·8-s + 1.40e4·9-s + 5.17e4·10-s − 2.16e3·11-s − 8.43e4·12-s − 2.85e4·13-s − 7.48e4·14-s + 3.05e5·15-s − 2.86e5·16-s + 3.29e4·17-s − 4.38e5·18-s − 4.48e5·19-s − 7.62e5·20-s − 4.41e5·21-s + 6.75e4·22-s − 7.08e5·23-s − 3.03e5·24-s + 8.05e5·25-s + 8.89e5·26-s + 1.03e6·27-s + 1.10e6·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 1.30·3-s + 0.896·4-s − 1.18·5-s + 1.80·6-s + 0.377·7-s + 0.142·8-s + 0.714·9-s + 1.63·10-s − 0.0446·11-s − 1.17·12-s − 0.277·13-s − 0.520·14-s + 1.55·15-s − 1.09·16-s + 0.0957·17-s − 0.983·18-s − 0.788·19-s − 1.06·20-s − 0.494·21-s + 0.0615·22-s − 0.528·23-s − 0.186·24-s + 0.412·25-s + 0.381·26-s + 0.373·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.002425444727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002425444727\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 31.1T + 512T^{2} \) |
| 3 | \( 1 + 183.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.66e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.16e3T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.29e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.08e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.49e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.77e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.82e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.58e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.19e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.83e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.45e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.05e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.52e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.70e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.20e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.48e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.04e9T + 7.60e17T^{2} \) |
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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
−11.68130909402451339757137871816, −11.14708386161236196830390035117, −10.27168644941441567407506643967, −8.887000811107613624581280239539, −7.81157520216842218847451342452, −6.95121423425670212864524460262, −5.38407485475970905468232002821, −4.04211487532968550884528993628, −1.66059908064621886440916643509, −0.03700038668019936861904113773,
0.03700038668019936861904113773, 1.66059908064621886440916643509, 4.04211487532968550884528993628, 5.38407485475970905468232002821, 6.95121423425670212864524460262, 7.81157520216842218847451342452, 8.887000811107613624581280239539, 10.27168644941441567407506643967, 11.14708386161236196830390035117, 11.68130909402451339757137871816
