| L(s) = 1 | − 272.·3-s − 1.00e4·7-s + 5.44e4·9-s − 4.70e4·11-s − 9.36e3·13-s − 1.08e5·17-s + 6.65e5·19-s + 2.72e6·21-s + 5.76e5·23-s − 9.45e6·27-s − 2.61e6·29-s − 3.87e6·31-s + 1.28e7·33-s − 1.41e7·37-s + 2.54e6·39-s + 4.62e6·41-s + 8.31e6·43-s + 2.51e7·47-s + 5.96e7·49-s + 2.95e7·51-s − 3.49e7·53-s − 1.81e8·57-s − 6.71e6·59-s − 4.75e6·61-s − 5.44e8·63-s − 1.38e8·67-s − 1.56e8·69-s + ⋯ |
| L(s) = 1 | − 1.94·3-s − 1.57·7-s + 2.76·9-s − 0.969·11-s − 0.0909·13-s − 0.315·17-s + 1.17·19-s + 3.05·21-s + 0.429·23-s − 3.42·27-s − 0.685·29-s − 0.754·31-s + 1.88·33-s − 1.24·37-s + 0.176·39-s + 0.255·41-s + 0.370·43-s + 0.753·47-s + 1.47·49-s + 0.611·51-s − 0.608·53-s − 2.27·57-s − 0.0721·59-s − 0.0440·61-s − 4.35·63-s − 0.841·67-s − 0.833·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.05553516229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05553516229\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 272.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 1.00e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.70e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.36e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.08e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.76e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.61e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.87e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.62e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.31e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.49e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.71e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.75e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.38e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.41e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.61e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.24e9T + 7.60e17T^{2} \) |
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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
−10.02663419888153148252671685739, −9.226154901948499701198291564708, −7.39466421312621383340028536555, −6.88981071525269349098308602931, −5.81579149360613389677703140018, −5.38952620619434848420333620790, −4.19312872822672472049750781959, −2.98237377185462861462000306138, −1.35916237034376316496678085715, −0.11555196789383632811764906073,
0.11555196789383632811764906073, 1.35916237034376316496678085715, 2.98237377185462861462000306138, 4.19312872822672472049750781959, 5.38952620619434848420333620790, 5.81579149360613389677703140018, 6.88981071525269349098308602931, 7.39466421312621383340028536555, 9.226154901948499701198291564708, 10.02663419888153148252671685739
