| L(s) = 1 | − 35.0·7-s − 25.6·11-s + 37.6·13-s + 95.7·17-s + 50.8·19-s − 110.·23-s + 54.5·29-s + 198.·31-s + 266.·37-s − 103.·41-s + 108·43-s − 597.·47-s + 887.·49-s + 305.·53-s + 223.·59-s + 485.·61-s − 876.·67-s − 585.·71-s − 1.13e3·73-s + 899.·77-s + 685.·79-s − 305.·83-s − 887.·89-s − 1.32e3·91-s − 556.·97-s − 1.59e3·101-s + 1.35e3·103-s + ⋯ |
| L(s) = 1 | − 1.89·7-s − 0.702·11-s + 0.803·13-s + 1.36·17-s + 0.614·19-s − 1.00·23-s + 0.349·29-s + 1.14·31-s + 1.18·37-s − 0.395·41-s + 0.383·43-s − 1.85·47-s + 2.58·49-s + 0.792·53-s + 0.493·59-s + 1.01·61-s − 1.59·67-s − 0.978·71-s − 1.82·73-s + 1.33·77-s + 0.975·79-s − 0.404·83-s − 1.05·89-s − 1.52·91-s − 0.582·97-s − 1.56·101-s + 1.29·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 35.0T + 343T^{2} \) |
| 11 | \( 1 + 25.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 103.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108T + 7.95e4T^{2} \) |
| 47 | \( 1 + 597.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 223.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 585.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 685.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 305.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 887.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 556.T + 9.12e5T^{2} \) |
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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
−8.482588798817288922471141938314, −7.76273384021937096837437608005, −6.83598371135290849480732919468, −6.07126520976805833662730436877, −5.54002677803800362322659932465, −4.20046531793093489851181107178, −3.28440684859641853450465863356, −2.73515759500030972777204040461, −1.11393513156581291718762437516, 0,
1.11393513156581291718762437516, 2.73515759500030972777204040461, 3.28440684859641853450465863356, 4.20046531793093489851181107178, 5.54002677803800362322659932465, 6.07126520976805833662730436877, 6.83598371135290849480732919468, 7.76273384021937096837437608005, 8.482588798817288922471141938314
