L(s) = 1 | − 2·4-s + 4·11-s + 3·13-s + 4·16-s − 3·17-s − 7·19-s − 23-s − 5·25-s + 29-s + 2·31-s + 37-s + 6·41-s − 11·43-s − 8·44-s − 10·47-s − 7·49-s − 6·52-s + 6·53-s + 11·59-s + 14·61-s − 8·64-s + 6·68-s + 15·71-s − 12·73-s + 14·76-s + 79-s + 2·83-s + ⋯ |
L(s) = 1 | − 4-s + 1.20·11-s + 0.832·13-s + 16-s − 0.727·17-s − 1.60·19-s − 0.208·23-s − 25-s + 0.185·29-s + 0.359·31-s + 0.164·37-s + 0.937·41-s − 1.67·43-s − 1.20·44-s − 1.45·47-s − 49-s − 0.832·52-s + 0.824·53-s + 1.43·59-s + 1.79·61-s − 64-s + 0.727·68-s + 1.78·71-s − 1.40·73-s + 1.60·76-s + 0.112·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.072951861163183414204383866975, −6.76777130438189262938336518541, −6.41098223414322409427369702225, −5.58762817816198073910432983640, −4.66003404692236358902664260720, −4.01933909460077403178405683484, −3.58255753087140901376892284759, −2.21796908624490588502252820857, −1.23855514836103712898121634803, 0,
1.23855514836103712898121634803, 2.21796908624490588502252820857, 3.58255753087140901376892284759, 4.01933909460077403178405683484, 4.66003404692236358902664260720, 5.58762817816198073910432983640, 6.41098223414322409427369702225, 6.76777130438189262938336518541, 8.072951861163183414204383866975