| L(s) = 1 | + 3-s − 1.80·5-s + 4.10·7-s + 9-s + 2.59·11-s − 3.46·13-s − 1.80·15-s + 1.68·17-s + 2.49·19-s + 4.10·21-s + 8.91·23-s − 1.74·25-s + 27-s − 4.42·29-s + 2.95·31-s + 2.59·33-s − 7.40·35-s + 0.918·37-s − 3.46·39-s − 1.83·41-s − 0.850·43-s − 1.80·45-s + 6.74·47-s + 9.88·49-s + 1.68·51-s + 4.65·53-s − 4.67·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.806·5-s + 1.55·7-s + 0.333·9-s + 0.781·11-s − 0.961·13-s − 0.465·15-s + 0.408·17-s + 0.573·19-s + 0.896·21-s + 1.85·23-s − 0.349·25-s + 0.192·27-s − 0.821·29-s + 0.529·31-s + 0.451·33-s − 1.25·35-s + 0.150·37-s − 0.555·39-s − 0.287·41-s − 0.129·43-s − 0.268·45-s + 0.983·47-s + 1.41·49-s + 0.235·51-s + 0.639·53-s − 0.630·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.986872389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.986872389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.80T + 5T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 - 2.59T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 1.68T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 - 8.91T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 - 0.918T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 0.850T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 - 4.65T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 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
−7.69822243112553543900798311369, −7.45579150973741705656542463726, −6.72746457485990525995970968682, −5.51977582631968101610209195117, −4.91480492775784136312209740454, −4.30120205972611558997069408020, −3.56541947932289919452942548475, −2.69610479640718123079042351509, −1.73478722833450916906733578568, −0.885872249558148497873950069949,
0.885872249558148497873950069949, 1.73478722833450916906733578568, 2.69610479640718123079042351509, 3.56541947932289919452942548475, 4.30120205972611558997069408020, 4.91480492775784136312209740454, 5.51977582631968101610209195117, 6.72746457485990525995970968682, 7.45579150973741705656542463726, 7.69822243112553543900798311369
