| L(s) = 1 | + 2-s − 2.90·3-s + 4-s + 1.52·5-s − 2.90·6-s − 7-s + 8-s + 5.42·9-s + 1.52·10-s + 5.80·11-s − 2.90·12-s − 14-s − 4.42·15-s + 16-s + 0.903·17-s + 5.42·18-s − 6.42·19-s + 1.52·20-s + 2.90·21-s + 5.80·22-s − 2.90·24-s − 2.67·25-s − 7.05·27-s − 28-s + 0.622·29-s − 4.42·30-s − 3.95·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.67·3-s + 0.5·4-s + 0.682·5-s − 1.18·6-s − 0.377·7-s + 0.353·8-s + 1.80·9-s + 0.482·10-s + 1.75·11-s − 0.838·12-s − 0.267·14-s − 1.14·15-s + 0.250·16-s + 0.219·17-s + 1.27·18-s − 1.47·19-s + 0.341·20-s + 0.633·21-s + 1.23·22-s − 0.592·24-s − 0.534·25-s − 1.35·27-s − 0.188·28-s + 0.115·29-s − 0.808·30-s − 0.710·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.903T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 29 | \( 1 - 0.622T + 29T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.57T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−6.87168376685624287740164641486, −6.62599483329784069847766452048, −6.22781699838624360307673693818, −5.47023382766370350104990251593, −4.90804057746184826747976722695, −4.07221040239929000023867036820, −3.48439811852134867917411840006, −1.99137839410306399288828194839, −1.37035672813975733090912511742, 0,
1.37035672813975733090912511742, 1.99137839410306399288828194839, 3.48439811852134867917411840006, 4.07221040239929000023867036820, 4.90804057746184826747976722695, 5.47023382766370350104990251593, 6.22781699838624360307673693818, 6.62599483329784069847766452048, 6.87168376685624287740164641486
