| L(s) = 1 | − 3-s + 1.31·7-s + 9-s + 1.55·11-s − 2.42·13-s + 1.31·17-s − 1.02·19-s − 1.31·21-s − 5.00·23-s − 27-s − 5.90·29-s + 8.85·31-s − 1.55·33-s − 5.02·37-s + 2.42·39-s + 7.95·41-s + 6.84·43-s − 7.75·47-s − 5.26·49-s − 1.31·51-s + 3.95·53-s + 1.02·57-s − 7.85·59-s − 3.50·61-s + 1.31·63-s + 8.33·67-s + 5.00·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.498·7-s + 0.333·9-s + 0.468·11-s − 0.673·13-s + 0.318·17-s − 0.234·19-s − 0.287·21-s − 1.04·23-s − 0.192·27-s − 1.09·29-s + 1.58·31-s − 0.270·33-s − 0.826·37-s + 0.388·39-s + 1.24·41-s + 1.04·43-s − 1.13·47-s − 0.751·49-s − 0.184·51-s + 0.542·53-s + 0.135·57-s − 1.02·59-s − 0.448·61-s + 0.166·63-s + 1.01·67-s + 0.602·69-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 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 - 6.84T + 43T^{2} \) |
| 47 | \( 1 + 7.75T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 - 8.33T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 8.20T + 83T^{2} \) |
| 89 | \( 1 - 9.57T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
| 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
−7.56805424647899930828297502756, −6.82403363951171242008907586200, −6.06223883486231333020465586678, −5.53900962760356375575513863614, −4.60895018598190039457847227647, −4.19724041565532098848170669767, −3.13615746965532494474521250157, −2.12165539845839858144172824578, −1.25936280066905935398397484754, 0,
1.25936280066905935398397484754, 2.12165539845839858144172824578, 3.13615746965532494474521250157, 4.19724041565532098848170669767, 4.60895018598190039457847227647, 5.53900962760356375575513863614, 6.06223883486231333020465586678, 6.82403363951171242008907586200, 7.56805424647899930828297502756
