| L(s) = 1 | + 3-s − 2.82·7-s + 9-s + 0.393·11-s − 13-s + 6.21·17-s − 7.34·19-s − 2.82·21-s + 1.44·23-s + 27-s − 0.828·29-s + 1.65·31-s + 0.393·33-s − 6.78·37-s − 39-s + 3.77·41-s + 2.51·43-s − 3.00·47-s + 1.00·49-s + 6.21·51-s + 9.55·53-s − 7.34·57-s + 0.393·59-s + 11.0·61-s − 2.82·63-s − 15.5·67-s + 1.44·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.06·7-s + 0.333·9-s + 0.118·11-s − 0.277·13-s + 1.50·17-s − 1.68·19-s − 0.617·21-s + 0.301·23-s + 0.192·27-s − 0.153·29-s + 0.297·31-s + 0.0684·33-s − 1.11·37-s − 0.160·39-s + 0.590·41-s + 0.383·43-s − 0.438·47-s + 0.142·49-s + 0.869·51-s + 1.31·53-s − 0.972·57-s + 0.0511·59-s + 1.41·61-s − 0.356·63-s − 1.90·67-s + 0.173·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.393T + 11T^{2} \) |
| 17 | \( 1 - 6.21T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 + 6.78T + 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 2.51T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 - 0.393T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 9.67T + 89T^{2} \) |
| 97 | \( 1 + 0.0418T + 97T^{2} \) |
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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
−7.38983932414227544302985332787, −6.96305565037860625337245143642, −6.12820333585418415598161679195, −5.55511650876490026472047832885, −4.50966629018661867398739041747, −3.80850160709359404240212361315, −3.10205004050751466307052271770, −2.41349048781533100077599982912, −1.32162337114402594317905037787, 0,
1.32162337114402594317905037787, 2.41349048781533100077599982912, 3.10205004050751466307052271770, 3.80850160709359404240212361315, 4.50966629018661867398739041747, 5.55511650876490026472047832885, 6.12820333585418415598161679195, 6.96305565037860625337245143642, 7.38983932414227544302985332787
