| L(s) = 1 | − 268.·3-s + 637.·7-s + 5.24e4·9-s + 4.90e4·11-s − 7.27e4·13-s − 6.73e4·17-s − 3.41e5·19-s − 1.71e5·21-s − 1.34e5·23-s − 8.81e6·27-s + 4.45e6·29-s − 4.56e5·31-s − 1.31e7·33-s − 1.30e7·37-s + 1.95e7·39-s − 2.56e7·41-s + 3.42e6·43-s − 3.39e7·47-s − 3.99e7·49-s + 1.80e7·51-s + 8.42e7·53-s + 9.16e7·57-s + 7.46e7·59-s + 1.78e8·61-s + 3.34e7·63-s + 6.94e7·67-s + 3.60e7·69-s + ⋯ |
| L(s) = 1 | − 1.91·3-s + 0.100·7-s + 2.66·9-s + 1.00·11-s − 0.706·13-s − 0.195·17-s − 0.600·19-s − 0.192·21-s − 0.100·23-s − 3.19·27-s + 1.17·29-s − 0.0887·31-s − 1.93·33-s − 1.14·37-s + 1.35·39-s − 1.41·41-s + 0.152·43-s − 1.01·47-s − 0.989·49-s + 0.374·51-s + 1.46·53-s + 1.15·57-s + 0.801·59-s + 1.64·61-s + 0.267·63-s + 0.420·67-s + 0.191·69-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 268.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 637.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.27e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.73e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.34e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.45e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.56e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.30e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.42e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.39e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.46e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.78e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.94e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.50e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.82e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.85e8T + 7.60e17T^{2} \) |
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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
−9.679086038751765723811782995833, −8.346781262949214263032713438068, −6.84721939836532501575337329560, −6.66032113257674828790778057928, −5.46045456486930213646119293111, −4.75973405008141794157062305000, −3.81220822044965051013463761184, −1.94260534857480988070618899603, −0.923339878482621828947555191737, 0,
0.923339878482621828947555191737, 1.94260534857480988070618899603, 3.81220822044965051013463761184, 4.75973405008141794157062305000, 5.46045456486930213646119293111, 6.66032113257674828790778057928, 6.84721939836532501575337329560, 8.346781262949214263032713438068, 9.679086038751765723811782995833
