| L(s) = 1 | + 23.7·2-s + 158.·3-s + 53.5·4-s + 2.24e3·5-s + 3.77e3·6-s + 8.56e3·7-s − 1.09e4·8-s + 5.45e3·9-s + 5.33e4·10-s − 4.17e3·11-s + 8.48e3·12-s + 1.00e5·13-s + 2.03e5·14-s + 3.55e5·15-s − 2.86e5·16-s + 1.37e5·17-s + 1.29e5·18-s − 554.·19-s + 1.20e5·20-s + 1.35e6·21-s − 9.91e4·22-s − 2.98e5·23-s − 1.72e6·24-s + 3.08e6·25-s + 2.38e6·26-s − 2.25e6·27-s + 4.58e5·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 1.13·3-s + 0.104·4-s + 1.60·5-s + 1.18·6-s + 1.34·7-s − 0.941·8-s + 0.277·9-s + 1.68·10-s − 0.0858·11-s + 0.118·12-s + 0.972·13-s + 1.41·14-s + 1.81·15-s − 1.09·16-s + 0.399·17-s + 0.291·18-s − 0.000976·19-s + 0.167·20-s + 1.52·21-s − 0.0902·22-s − 0.222·23-s − 1.06·24-s + 1.57·25-s + 1.02·26-s − 0.816·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(8.595072729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.595072729\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 1.50e9T \) |
| good | 2 | \( 1 - 23.7T + 512T^{2} \) |
| 3 | \( 1 - 158.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.24e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.17e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.00e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.37e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 554.T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.98e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.69e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.33e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.29e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.16e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.29e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.52e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.63e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.09e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.38e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.16e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.05e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.95e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.62e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.30e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.83e8T + 7.60e17T^{2} \) |
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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
−10.86011147449732297507170876500, −9.674286777870539350474874675281, −8.797837516976353546792318437388, −8.081612989077519577368198368053, −6.35594321735404241578829856193, −5.48438462040836727896024250067, −4.55563381863726225202810900959, −3.24392026065528075686075103971, −2.28559692105422521530346985980, −1.32145789103419359426563614402,
1.32145789103419359426563614402, 2.28559692105422521530346985980, 3.24392026065528075686075103971, 4.55563381863726225202810900959, 5.48438462040836727896024250067, 6.35594321735404241578829856193, 8.081612989077519577368198368053, 8.797837516976353546792318437388, 9.674286777870539350474874675281, 10.86011147449732297507170876500
