| L(s) = 1 | − 3-s − 3.80·7-s + 9-s + 0.190·11-s − 1.67·13-s + 4.60·17-s − 2.64·19-s + 3.80·21-s + 6.35·23-s − 27-s + 2.52·29-s + 3.74·31-s − 0.190·33-s − 11.8·37-s + 1.67·39-s − 7.18·41-s − 9.22·43-s − 4.54·47-s + 7.51·49-s − 4.60·51-s + 9.43·53-s + 2.64·57-s + 7.14·59-s + 9.53·61-s − 3.80·63-s + 6.05·67-s − 6.35·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.44·7-s + 0.333·9-s + 0.0575·11-s − 0.465·13-s + 1.11·17-s − 0.605·19-s + 0.831·21-s + 1.32·23-s − 0.192·27-s + 0.467·29-s + 0.673·31-s − 0.0332·33-s − 1.95·37-s + 0.268·39-s − 1.12·41-s − 1.40·43-s − 0.663·47-s + 1.07·49-s − 0.644·51-s + 1.29·53-s + 0.349·57-s + 0.929·59-s + 1.22·61-s − 0.480·63-s + 0.739·67-s − 0.765·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 - 0.190T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 + 9.22T + 43T^{2} \) |
| 47 | \( 1 + 4.54T + 47T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 - 7.14T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 - 6.05T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 5.21T + 73T^{2} \) |
| 79 | \( 1 + 3.11T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
| 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
−7.27286601248051127468361272522, −6.72055061013184099564747145242, −6.36880017518034548873463643684, −5.28183459744892954194050053339, −5.01413624640324418685659956941, −3.72241599014153807443303038233, −3.30416443954371052308091277185, −2.31670214870625656418758971922, −1.05220625648217339435955136575, 0,
1.05220625648217339435955136575, 2.31670214870625656418758971922, 3.30416443954371052308091277185, 3.72241599014153807443303038233, 5.01413624640324418685659956941, 5.28183459744892954194050053339, 6.36880017518034548873463643684, 6.72055061013184099564747145242, 7.27286601248051127468361272522
