| L(s) = 1 | − 2-s + 1.41·3-s + 4-s + 4.16·5-s − 1.41·6-s − 8-s − 0.999·9-s − 4.16·10-s − 6.36·11-s + 1.41·12-s − 4.16·13-s + 5.88·15-s + 16-s − 6.25·17-s + 0.999·18-s − 0.737·19-s + 4.16·20-s + 6.36·22-s − 5.52·23-s − 1.41·24-s + 12.3·25-s + 4.16·26-s − 5.65·27-s − 29-s − 5.88·30-s + 5.51·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s + 1.86·5-s − 0.577·6-s − 0.353·8-s − 0.333·9-s − 1.31·10-s − 1.91·11-s + 0.408·12-s − 1.15·13-s + 1.51·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s − 0.169·19-s + 0.930·20-s + 1.35·22-s − 1.15·23-s − 0.288·24-s + 2.46·25-s + 0.816·26-s − 1.08·27-s − 0.185·29-s − 1.07·30-s + 0.990·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 + 6.36T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 + 6.25T + 17T^{2} \) |
| 19 | \( 1 + 0.737T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 5.04T + 53T^{2} \) |
| 59 | \( 1 - 3.42T + 59T^{2} \) |
| 61 | \( 1 - 3.50T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 4.56T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 + 6.84T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.412831749707089165582907321297, −7.972002262886923464393140897146, −6.95490958267118526526363290626, −6.18138561653107222440190388862, −5.42271385436880019993045557106, −4.69042689392332505033868951061, −2.94328950736967876146924692180, −2.38183526413770336734782166320, −1.97384073207015165481187856345, 0,
1.97384073207015165481187856345, 2.38183526413770336734782166320, 2.94328950736967876146924692180, 4.69042689392332505033868951061, 5.42271385436880019993045557106, 6.18138561653107222440190388862, 6.95490958267118526526363290626, 7.972002262886923464393140897146, 8.412831749707089165582907321297
