| L(s) = 1 | − 1.45·2-s − 0.257·3-s + 0.107·4-s + 0.373·6-s − 4.95·7-s + 2.74·8-s − 2.93·9-s + 4.28·11-s − 0.0277·12-s − 0.0395·13-s + 7.19·14-s − 4.20·16-s + 2.35·17-s + 4.25·18-s − 0.622·19-s + 1.27·21-s − 6.21·22-s − 4.49·23-s − 0.706·24-s + 0.0574·26-s + 1.52·27-s − 0.534·28-s − 0.366·29-s + 5.71·31-s + 0.608·32-s − 1.10·33-s − 3.42·34-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 0.148·3-s + 0.0538·4-s + 0.152·6-s − 1.87·7-s + 0.971·8-s − 0.977·9-s + 1.29·11-s − 0.00799·12-s − 0.0109·13-s + 1.92·14-s − 1.05·16-s + 0.572·17-s + 1.00·18-s − 0.142·19-s + 0.278·21-s − 1.32·22-s − 0.936·23-s − 0.144·24-s + 0.0112·26-s + 0.293·27-s − 0.100·28-s − 0.0680·29-s + 1.02·31-s + 0.107·32-s − 0.191·33-s − 0.587·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 + 0.257T + 3T^{2} \) |
| 7 | \( 1 + 4.95T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 + 0.0395T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 0.622T + 19T^{2} \) |
| 23 | \( 1 + 4.49T + 23T^{2} \) |
| 29 | \( 1 + 0.366T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 + 0.447T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 6.71T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 + 0.622T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 4.20T + 97T^{2} \) |
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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
−8.211064170016204978228269775636, −7.13607846220279840437755253803, −6.52167227055610029015813576373, −6.03784204055688635208785168469, −5.03924159900832658865776898529, −3.88017436101905597588469297139, −3.38245898903025875422849534995, −2.26713238944527245523935240044, −0.943082889569808157856266105639, 0,
0.943082889569808157856266105639, 2.26713238944527245523935240044, 3.38245898903025875422849534995, 3.88017436101905597588469297139, 5.03924159900832658865776898529, 6.03784204055688635208785168469, 6.52167227055610029015813576373, 7.13607846220279840437755253803, 8.211064170016204978228269775636
