| L(s) = 1 | + 2.64·3-s + 1.64·5-s − 3.64·7-s + 4.00·9-s − 13-s + 4.35·15-s − 3·17-s − 3.64·19-s − 9.64·21-s − 1.35·23-s − 2.29·25-s + 2.64·27-s − 3·29-s + 5.29·31-s − 6·35-s − 8.93·37-s − 2.64·39-s − 1.64·41-s − 3.93·43-s + 6.58·45-s − 7.64·47-s + 6.29·49-s − 7.93·51-s − 3·53-s − 9.64·57-s − 3.29·59-s + 3.70·61-s + ⋯ |
| L(s) = 1 | + 1.52·3-s + 0.736·5-s − 1.37·7-s + 1.33·9-s − 0.277·13-s + 1.12·15-s − 0.727·17-s − 0.836·19-s − 2.10·21-s − 0.282·23-s − 0.458·25-s + 0.509·27-s − 0.557·29-s + 0.950·31-s − 1.01·35-s − 1.46·37-s − 0.423·39-s − 0.257·41-s − 0.600·43-s + 0.981·45-s − 1.11·47-s + 0.898·49-s − 1.11·51-s − 0.412·53-s − 1.27·57-s − 0.428·59-s + 0.474·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 3.64T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 + 1.64T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 + 7.64T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.904789342168295218178218482690, −6.81907090650934438405940697030, −6.56284675377623543776155628951, −5.66116484556508456805787770624, −4.62059752641781387539687770025, −3.71914632977830176612740609951, −3.17097494147735922408954605348, −2.34935697043690121428751635147, −1.78172682962305286231771694459, 0,
1.78172682962305286231771694459, 2.34935697043690121428751635147, 3.17097494147735922408954605348, 3.71914632977830176612740609951, 4.62059752641781387539687770025, 5.66116484556508456805787770624, 6.56284675377623543776155628951, 6.81907090650934438405940697030, 7.904789342168295218178218482690
