| L(s) = 1 | + 1.65·3-s − 0.641·5-s + 1.04·7-s − 0.251·9-s + 3.12·11-s − 1.00·13-s − 1.06·15-s − 4.94·17-s + 8.16·19-s + 1.73·21-s − 1.05·23-s − 4.58·25-s − 5.39·27-s − 10.2·29-s − 7.21·31-s + 5.18·33-s − 0.672·35-s − 2.17·37-s − 1.66·39-s + 9.09·41-s − 1.16·43-s + 0.161·45-s − 3.67·47-s − 5.90·49-s − 8.20·51-s − 5.62·53-s − 2.00·55-s + ⋯ |
| L(s) = 1 | + 0.957·3-s − 0.287·5-s + 0.396·7-s − 0.0837·9-s + 0.943·11-s − 0.278·13-s − 0.274·15-s − 1.20·17-s + 1.87·19-s + 0.379·21-s − 0.220·23-s − 0.917·25-s − 1.03·27-s − 1.90·29-s − 1.29·31-s + 0.903·33-s − 0.113·35-s − 0.357·37-s − 0.266·39-s + 1.42·41-s − 0.177·43-s + 0.0240·45-s − 0.536·47-s − 0.843·49-s − 1.14·51-s − 0.772·53-s − 0.270·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 + 0.641T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 - 8.16T + 19T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 - 9.09T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 + 5.62T + 53T^{2} \) |
| 59 | \( 1 + 9.72T + 59T^{2} \) |
| 61 | \( 1 - 0.384T + 61T^{2} \) |
| 67 | \( 1 - 6.44T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.55174402035737630066011747247, −7.09772414636662222346528976847, −6.04381542791805405306396683052, −5.42726729735183583969924806510, −4.49469236210194633297541197967, −3.72308779361037714392105524987, −3.24847386108181383900611313261, −2.17637974305909546599266043276, −1.54291700175339548614996248108, 0,
1.54291700175339548614996248108, 2.17637974305909546599266043276, 3.24847386108181383900611313261, 3.72308779361037714392105524987, 4.49469236210194633297541197967, 5.42726729735183583969924806510, 6.04381542791805405306396683052, 7.09772414636662222346528976847, 7.55174402035737630066011747247
