| L(s) = 1 | + 2.60·7-s − 3·9-s + 6.60·11-s − 7.21·17-s + 1.39·19-s − 5·25-s − 7.21·29-s − 10.6·31-s − 5.39·47-s − 0.211·49-s − 2·53-s − 11.8·59-s + 6·61-s − 7.81·63-s − 14.6·67-s + 15.8·71-s + 17.2·77-s + 9·81-s + 3.81·83-s − 19.8·99-s − 14·101-s + 7.21·113-s − 18.7·119-s + ⋯ |
| L(s) = 1 | + 0.984·7-s − 9-s + 1.99·11-s − 1.74·17-s + 0.319·19-s − 25-s − 1.33·29-s − 1.90·31-s − 0.786·47-s − 0.0301·49-s − 0.274·53-s − 1.53·59-s + 0.768·61-s − 0.984·63-s − 1.78·67-s + 1.87·71-s + 1.96·77-s + 81-s + 0.418·83-s − 1.99·99-s − 1.39·101-s + 0.678·113-s − 1.72·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 - 6.60T + 11T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.85657080605858470662776653455, −7.08839238731151107299297754803, −6.35464806187057239310107954085, −5.71524195192442254381361816615, −4.85840413614407504474505684834, −4.05946404167772737791609805592, −3.45343090505247130253682627467, −2.10866697353315047337281727212, −1.55926025733303843746617125086, 0,
1.55926025733303843746617125086, 2.10866697353315047337281727212, 3.45343090505247130253682627467, 4.05946404167772737791609805592, 4.85840413614407504474505684834, 5.71524195192442254381361816615, 6.35464806187057239310107954085, 7.08839238731151107299297754803, 7.85657080605858470662776653455
