| L(s) = 1 | − 1.09·2-s + 3-s − 0.803·4-s − 1.09·6-s − 1.13·7-s + 3.06·8-s + 9-s − 0.803·12-s + 5.16·13-s + 1.24·14-s − 1.74·16-s − 0.942·17-s − 1.09·18-s + 7.90·19-s − 1.13·21-s + 5.10·23-s + 3.06·24-s − 5.64·26-s + 27-s + 0.915·28-s − 4.85·29-s + 4.23·31-s − 4.22·32-s + 1.03·34-s − 0.803·36-s − 0.521·37-s − 8.64·38-s + ⋯ |
| L(s) = 1 | − 0.773·2-s + 0.577·3-s − 0.401·4-s − 0.446·6-s − 0.430·7-s + 1.08·8-s + 0.333·9-s − 0.232·12-s + 1.43·13-s + 0.332·14-s − 0.436·16-s − 0.228·17-s − 0.257·18-s + 1.81·19-s − 0.248·21-s + 1.06·23-s + 0.625·24-s − 1.10·26-s + 0.192·27-s + 0.172·28-s − 0.901·29-s + 0.760·31-s − 0.746·32-s + 0.176·34-s − 0.133·36-s − 0.0858·37-s − 1.40·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724954873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724954873\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 + 0.942T + 17T^{2} \) |
| 19 | \( 1 - 7.90T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 0.521T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 5.85T + 43T^{2} \) |
| 47 | \( 1 + 0.302T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.831T + 61T^{2} \) |
| 67 | \( 1 - 7.05T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 2.73T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.947096278920367331840610620129, −7.18194894882001376103274312219, −6.65150593161046192904560494948, −5.58863035137106623966157358960, −5.03511979290825271100447542098, −3.95580639537453201141923456488, −3.54006562359454372663432092454, −2.61442776198897631258262146151, −1.41865528958796829654820074672, −0.78796317335720383787200029942,
0.78796317335720383787200029942, 1.41865528958796829654820074672, 2.61442776198897631258262146151, 3.54006562359454372663432092454, 3.95580639537453201141923456488, 5.03511979290825271100447542098, 5.58863035137106623966157358960, 6.65150593161046192904560494948, 7.18194894882001376103274312219, 7.947096278920367331840610620129
