L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s − 1.41·6-s + 9-s + 1.41·11-s − 1.00·12-s + 13-s − 0.999·16-s + 1.41·18-s + 2.00·22-s + 1.41·26-s − 27-s − 1.41·32-s − 1.41·33-s + 1.00·36-s − 39-s − 1.41·41-s + 1.41·44-s − 1.41·47-s + 0.999·48-s + 49-s + 1.00·52-s − 1.41·54-s − 1.41·59-s − 1.00·64-s − 2.00·66-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s − 1.41·6-s + 9-s + 1.41·11-s − 1.00·12-s + 13-s − 0.999·16-s + 1.41·18-s + 2.00·22-s + 1.41·26-s − 27-s − 1.41·32-s − 1.41·33-s + 1.00·36-s − 39-s − 1.41·41-s + 1.41·44-s − 1.41·47-s + 0.999·48-s + 49-s + 1.00·52-s − 1.41·54-s − 1.41·59-s − 1.00·64-s − 2.00·66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.608732200\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.608732200\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.54240352641549579210268639796, −9.481904633147799879855693562761, −8.600009344692936330989033487723, −7.15886534428999144870341447523, −6.39520055325032331257789206695, −5.91624621140999306640974658010, −4.91298554642986538405636031710, −4.12805739311864584524678157781, −3.35946204443753001073807812280, −1.56816664368498350807741072749,
1.56816664368498350807741072749, 3.35946204443753001073807812280, 4.12805739311864584524678157781, 4.91298554642986538405636031710, 5.91624621140999306640974658010, 6.39520055325032331257789206695, 7.15886534428999144870341447523, 8.600009344692936330989033487723, 9.481904633147799879855693562761, 10.54240352641549579210268639796