| L(s) = 1 | + 70.3·3-s − 162.·5-s − 1.47e4·9-s + 9.08e4·11-s + 1.67e5·13-s − 1.14e4·15-s − 9.82e3·17-s − 4.75e5·19-s − 2.21e5·23-s − 1.92e6·25-s − 2.42e6·27-s + 8.20e5·29-s + 2.83e6·31-s + 6.39e6·33-s − 2.71e6·37-s + 1.18e7·39-s + 1.90e7·41-s + 2.58e7·43-s + 2.39e6·45-s − 3.51e7·47-s − 6.91e5·51-s − 8.20e7·53-s − 1.47e7·55-s − 3.34e7·57-s + 1.33e8·59-s + 1.01e8·61-s − 2.73e7·65-s + ⋯ |
| L(s) = 1 | + 0.501·3-s − 0.116·5-s − 0.748·9-s + 1.87·11-s + 1.63·13-s − 0.0584·15-s − 0.0285·17-s − 0.837·19-s − 0.164·23-s − 0.986·25-s − 0.877·27-s + 0.215·29-s + 0.550·31-s + 0.938·33-s − 0.237·37-s + 0.817·39-s + 1.05·41-s + 1.15·43-s + 0.0871·45-s − 1.04·47-s − 0.0143·51-s − 1.42·53-s − 0.217·55-s − 0.419·57-s + 1.43·59-s + 0.940·61-s − 0.189·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.910093451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.910093451\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 70.3T + 1.96e4T^{2} \) |
| 5 | \( 1 + 162.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 9.08e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.67e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.82e3T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.75e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.21e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 8.20e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.71e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.90e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.20e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.33e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.01e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.48e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.94e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.20e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.01e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.49e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.54e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.51e9T + 7.60e17T^{2} \) |
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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
−11.08219123226741039877227976646, −9.618010462741817334650560332794, −8.779235917420769330779376692119, −8.120060841387394623829471137717, −6.59339464797209476455859014839, −5.91668503861838351132163422064, −4.15501753610292549313406340126, −3.46687428748430495546454873271, −1.99280471055444261368370554944, −0.829251027335053799159110502979,
0.829251027335053799159110502979, 1.99280471055444261368370554944, 3.46687428748430495546454873271, 4.15501753610292549313406340126, 5.91668503861838351132163422064, 6.59339464797209476455859014839, 8.120060841387394623829471137717, 8.779235917420769330779376692119, 9.618010462741817334650560332794, 11.08219123226741039877227976646
