| L(s) = 1 | + 0.571·2-s + 3-s − 1.67·4-s + 0.571·6-s + 1.42·7-s − 2.10·8-s + 9-s + 3.24·11-s − 1.67·12-s + 13-s + 0.816·14-s + 2.14·16-s + 1.85·17-s + 0.571·18-s − 1.81·19-s + 1.42·21-s + 1.85·22-s − 1.52·23-s − 2.10·24-s + 0.571·26-s + 27-s − 2.38·28-s + 2.34·29-s + 6.38·31-s + 5.42·32-s + 3.24·33-s + 1.06·34-s + ⋯ |
| L(s) = 1 | + 0.404·2-s + 0.577·3-s − 0.836·4-s + 0.233·6-s + 0.539·7-s − 0.742·8-s + 0.333·9-s + 0.978·11-s − 0.482·12-s + 0.277·13-s + 0.218·14-s + 0.535·16-s + 0.450·17-s + 0.134·18-s − 0.416·19-s + 0.311·21-s + 0.395·22-s − 0.318·23-s − 0.428·24-s + 0.112·26-s + 0.192·27-s − 0.451·28-s + 0.435·29-s + 1.14·31-s + 0.959·32-s + 0.564·33-s + 0.182·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.173214094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173214094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.571T + 2T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.81T + 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 9.81T + 71T^{2} \) |
| 73 | \( 1 - 5.32T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.51T + 97T^{2} \) |
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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
−9.844721506773699111328716011180, −9.071165007575806544352561052606, −8.463476932824934229933285757593, −7.68225162045794193403571433804, −6.49177706065658420844082811486, −5.59733495935753183320243485615, −4.45146916406531925091292888584, −3.93778791930814537354389050064, −2.74625660114410623806951208006, −1.18199453170977942896055959668,
1.18199453170977942896055959668, 2.74625660114410623806951208006, 3.93778791930814537354389050064, 4.45146916406531925091292888584, 5.59733495935753183320243485615, 6.49177706065658420844082811486, 7.68225162045794193403571433804, 8.463476932824934229933285757593, 9.071165007575806544352561052606, 9.844721506773699111328716011180
