| L(s) = 1 | + 1.91·3-s + 1.16·5-s − 1.88·7-s + 0.648·9-s + 4.20·11-s − 6.01·13-s + 2.21·15-s + 2.39·17-s − 7.69·19-s − 3.59·21-s − 2.06·23-s − 3.65·25-s − 4.49·27-s + 4.33·29-s + 2.53·31-s + 8.04·33-s − 2.18·35-s + 8.55·37-s − 11.4·39-s − 1.28·41-s + 5.01·43-s + 0.752·45-s + 0.868·47-s − 3.46·49-s + 4.58·51-s − 3.75·53-s + 4.88·55-s + ⋯ |
| L(s) = 1 | + 1.10·3-s + 0.519·5-s − 0.710·7-s + 0.216·9-s + 1.26·11-s − 1.66·13-s + 0.572·15-s + 0.582·17-s − 1.76·19-s − 0.784·21-s − 0.431·23-s − 0.730·25-s − 0.864·27-s + 0.805·29-s + 0.454·31-s + 1.39·33-s − 0.368·35-s + 1.40·37-s − 1.83·39-s − 0.201·41-s + 0.764·43-s + 0.112·45-s + 0.126·47-s − 0.494·49-s + 0.641·51-s − 0.515·53-s + 0.658·55-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 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 7.69T + 19T^{2} \) |
| 23 | \( 1 + 2.06T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 - 0.868T + 47T^{2} \) |
| 53 | \( 1 + 3.75T + 53T^{2} \) |
| 59 | \( 1 + 5.59T + 59T^{2} \) |
| 61 | \( 1 - 7.36T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.64674080344940003669526541301, −6.69054170206175251649328241400, −6.30583636265944714636541686549, −5.47965219990632622158809558532, −4.35257237253957443283240346450, −3.94888793705546226501284757105, −2.83318754226337549340788281488, −2.47050114766297913781550471695, −1.53226753107925386192159298105, 0,
1.53226753107925386192159298105, 2.47050114766297913781550471695, 2.83318754226337549340788281488, 3.94888793705546226501284757105, 4.35257237253957443283240346450, 5.47965219990632622158809558532, 6.30583636265944714636541686549, 6.69054170206175251649328241400, 7.64674080344940003669526541301
