| L(s) = 1 | + 3-s + 5-s + 9-s − 0.828·11-s + 15-s + 2.82·17-s − 1.41·19-s + 3.41·23-s + 25-s + 27-s − 2·29-s + 5.41·31-s − 0.828·33-s + 3.17·37-s + 0.343·41-s + 5.65·43-s + 45-s + 0.828·47-s + 2.82·51-s + 0.585·53-s − 0.828·55-s − 1.41·57-s + 8.48·59-s − 7.07·61-s + 6.82·67-s + 3.41·69-s + 7.65·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.333·9-s − 0.249·11-s + 0.258·15-s + 0.685·17-s − 0.324·19-s + 0.711·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s + 0.972·31-s − 0.144·33-s + 0.521·37-s + 0.0535·41-s + 0.862·43-s + 0.149·45-s + 0.120·47-s + 0.396·51-s + 0.0804·53-s − 0.111·55-s − 0.187·57-s + 1.10·59-s − 0.905·61-s + 0.834·67-s + 0.411·69-s + 0.908·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.603068560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.603068560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 - 0.585T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 7.65T + 71T^{2} \) |
| 73 | \( 1 - 7.31T + 73T^{2} \) |
| 79 | \( 1 - 0.343T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.736157644184987464916460779661, −8.056122052668951191708299846578, −7.33050460349543467152309462735, −6.52129531530860932785780212040, −5.68553492033231893552946878884, −4.87450545681025207947455426289, −3.94372293153195936679480951836, −2.98905430318354695490693506275, −2.20641826841875470028697063644, −1.00316321075256245548018145412,
1.00316321075256245548018145412, 2.20641826841875470028697063644, 2.98905430318354695490693506275, 3.94372293153195936679480951836, 4.87450545681025207947455426289, 5.68553492033231893552946878884, 6.52129531530860932785780212040, 7.33050460349543467152309462735, 8.056122052668951191708299846578, 8.736157644184987464916460779661
