| L(s) = 1 | + 3.23·5-s + 7-s + 2·11-s + 4.47·13-s − 3.23·17-s − 19-s + 2·23-s + 5.47·25-s + 2.76·29-s + 4·31-s + 3.23·35-s − 4.47·37-s + 2.47·41-s − 1.52·43-s + 4.76·47-s + 49-s + 10.1·53-s + 6.47·55-s − 4.94·59-s − 8.47·61-s + 14.4·65-s − 12.9·67-s + 2.76·71-s + 6·73-s + 2·77-s + 0.944·79-s + 11.2·83-s + ⋯ |
| L(s) = 1 | + 1.44·5-s + 0.377·7-s + 0.603·11-s + 1.24·13-s − 0.784·17-s − 0.229·19-s + 0.417·23-s + 1.09·25-s + 0.513·29-s + 0.718·31-s + 0.546·35-s − 0.735·37-s + 0.386·41-s − 0.232·43-s + 0.694·47-s + 0.142·49-s + 1.39·53-s + 0.872·55-s − 0.643·59-s − 1.08·61-s + 1.79·65-s − 1.58·67-s + 0.328·71-s + 0.702·73-s + 0.227·77-s + 0.106·79-s + 1.23·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.148664097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.148664097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 0.944T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
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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
−8.561266748619750062654643145899, −7.52568362109191971729194594285, −6.49965704325248316658661055143, −6.27630192873524620341954861513, −5.44321374509681900649684391464, −4.65161518070291075287410499799, −3.78878996874181176894084618008, −2.71119375745888847995330347808, −1.84769743631374320526850239478, −1.07111329255288240712707202781,
1.07111329255288240712707202781, 1.84769743631374320526850239478, 2.71119375745888847995330347808, 3.78878996874181176894084618008, 4.65161518070291075287410499799, 5.44321374509681900649684391464, 6.27630192873524620341954861513, 6.49965704325248316658661055143, 7.52568362109191971729194594285, 8.561266748619750062654643145899
