| L(s) = 1 | − 2.69·2-s + 3-s + 5.27·4-s − 2.69·6-s − 3.56·7-s − 8.84·8-s + 9-s − 0.0695·11-s + 5.27·12-s + 4.85·13-s + 9.62·14-s + 13.2·16-s + 3.03·17-s − 2.69·18-s + 5.05·19-s − 3.56·21-s + 0.187·22-s − 7.43·23-s − 8.84·24-s − 13.1·26-s + 27-s − 18.8·28-s + 1.95·29-s − 5.63·31-s − 18.1·32-s − 0.0695·33-s − 8.18·34-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.577·3-s + 2.63·4-s − 1.10·6-s − 1.34·7-s − 3.12·8-s + 0.333·9-s − 0.0209·11-s + 1.52·12-s + 1.34·13-s + 2.57·14-s + 3.32·16-s + 0.735·17-s − 0.635·18-s + 1.15·19-s − 0.778·21-s + 0.0400·22-s − 1.54·23-s − 1.80·24-s − 2.56·26-s + 0.192·27-s − 3.55·28-s + 0.362·29-s − 1.01·31-s − 3.21·32-s − 0.0121·33-s − 1.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7744633214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7744633214\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 0.0695T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 + 5.63T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 - 0.244T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 - 8.37T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 - 7.25T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 + 8.80T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−9.238657297789705018553289050296, −8.561859987432612741852963473075, −7.87756504570947316864465747334, −7.18477370784794993465519466379, −6.34516935987851453453268901480, −5.76031499888530558231954372044, −3.66199388639239091490922935759, −3.10196961252874720425100359154, −1.92555570786812532569720401298, −0.76812721676493491923331458873,
0.76812721676493491923331458873, 1.92555570786812532569720401298, 3.10196961252874720425100359154, 3.66199388639239091490922935759, 5.76031499888530558231954372044, 6.34516935987851453453268901480, 7.18477370784794993465519466379, 7.87756504570947316864465747334, 8.561859987432612741852963473075, 9.238657297789705018553289050296
