| L(s) = 1 | + 670.·2-s + 6.56e3·3-s + 3.18e5·4-s + 4.39e6·6-s + 2.70e7·7-s + 1.25e8·8-s + 4.30e7·9-s + 1.01e9·11-s + 2.09e9·12-s + 1.04e9·13-s + 1.81e10·14-s + 4.25e10·16-s − 2.00e10·17-s + 2.88e10·18-s + 1.64e10·19-s + 1.77e11·21-s + 6.81e11·22-s − 5.92e11·23-s + 8.25e11·24-s + 7.03e11·26-s + 2.82e11·27-s + 8.62e12·28-s + 4.83e11·29-s − 5.20e12·31-s + 1.20e13·32-s + 6.66e12·33-s − 1.34e13·34-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.43·4-s + 1.06·6-s + 1.77·7-s + 2.65·8-s + 0.333·9-s + 1.42·11-s + 1.40·12-s + 0.356·13-s + 3.28·14-s + 2.47·16-s − 0.695·17-s + 0.617·18-s + 0.222·19-s + 1.02·21-s + 2.64·22-s − 1.57·23-s + 1.53·24-s + 0.660·26-s + 0.192·27-s + 4.31·28-s + 0.179·29-s − 1.09·31-s + 1.94·32-s + 0.825·33-s − 1.28·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(13.40130139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.40130139\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 670.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.70e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.01e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 1.04e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.00e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.64e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.92e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 4.83e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.20e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.82e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.56e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 5.83e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 7.43e12T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.01e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 3.44e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.43e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.19e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 5.42e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 5.84e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.03e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.18e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.43e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.06e17T + 5.95e33T^{2} \) |
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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
−11.62728756298988837244490034341, −10.68427629909040830901966899299, −8.788458456868409905088852611455, −7.59921030423892801499846403639, −6.48828860711742732486958864177, −5.26609969682928057766852241236, −4.28420635963863965014546176718, −3.61073015452786595191169705203, −2.04007275454055922300651269508, −1.53899112123255514038726808407,
1.53899112123255514038726808407, 2.04007275454055922300651269508, 3.61073015452786595191169705203, 4.28420635963863965014546176718, 5.26609969682928057766852241236, 6.48828860711742732486958864177, 7.59921030423892801499846403639, 8.788458456868409905088852611455, 10.68427629909040830901966899299, 11.62728756298988837244490034341
