| L(s) = 1 | − 2.05·3-s + 3.84·5-s − 4.00·7-s + 1.23·9-s + 3.28·11-s − 3.96·13-s − 7.92·15-s + 0.871·17-s − 3.06·19-s + 8.23·21-s − 1.95·23-s + 9.82·25-s + 3.62·27-s + 0.855·29-s − 6.85·31-s − 6.75·33-s − 15.4·35-s − 1.69·37-s + 8.15·39-s + 6.10·41-s + 2.26·43-s + 4.76·45-s + 3.98·47-s + 9.00·49-s − 1.79·51-s − 0.873·53-s + 12.6·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s + 1.72·5-s − 1.51·7-s + 0.412·9-s + 0.989·11-s − 1.09·13-s − 2.04·15-s + 0.211·17-s − 0.702·19-s + 1.79·21-s − 0.406·23-s + 1.96·25-s + 0.698·27-s + 0.158·29-s − 1.23·31-s − 1.17·33-s − 2.60·35-s − 0.279·37-s + 1.30·39-s + 0.953·41-s + 0.345·43-s + 0.710·45-s + 0.580·47-s + 1.28·49-s − 0.251·51-s − 0.120·53-s + 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.161928263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161928263\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.05T + 3T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 - 3.28T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 - 0.871T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 1.95T + 23T^{2} \) |
| 29 | \( 1 - 0.855T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 - 6.10T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 0.873T + 53T^{2} \) |
| 59 | \( 1 - 6.66T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 + 0.0135T + 79T^{2} \) |
| 83 | \( 1 + 2.36T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−8.843362886287015775602591817407, −7.30287307276134976846888893063, −6.65414049314774266357742014215, −6.16376751414788235344682007570, −5.71133566299453007006729475899, −4.99224785709278897131684829923, −3.91852994922317067619013232671, −2.78837183924884353656591337941, −1.94713625740210885006006919475, −0.63804201441077367251047592037,
0.63804201441077367251047592037, 1.94713625740210885006006919475, 2.78837183924884353656591337941, 3.91852994922317067619013232671, 4.99224785709278897131684829923, 5.71133566299453007006729475899, 6.16376751414788235344682007570, 6.65414049314774266357742014215, 7.30287307276134976846888893063, 8.843362886287015775602591817407
