| L(s) = 1 | + 42.0·2-s + 1.25e3·4-s − 1.40e3·5-s − 5.13e3·7-s + 3.13e4·8-s − 5.91e4·10-s − 2.65e4·11-s + 7.14e4·13-s − 2.16e5·14-s + 6.74e5·16-s + 8.35e4·17-s − 5.48e5·19-s − 1.76e6·20-s − 1.11e6·22-s − 1.15e6·23-s + 2.72e4·25-s + 3.00e6·26-s − 6.46e6·28-s − 1.44e6·29-s − 6.05e6·31-s + 1.23e7·32-s + 3.51e6·34-s + 7.23e6·35-s − 9.50e6·37-s − 2.30e7·38-s − 4.41e7·40-s − 1.75e7·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.45·4-s − 1.00·5-s − 0.808·7-s + 2.70·8-s − 1.87·10-s − 0.546·11-s + 0.693·13-s − 1.50·14-s + 2.57·16-s + 0.242·17-s − 0.965·19-s − 2.47·20-s − 1.01·22-s − 0.864·23-s + 0.0139·25-s + 1.28·26-s − 1.98·28-s − 0.378·29-s − 1.17·31-s + 2.07·32-s + 0.450·34-s + 0.814·35-s − 0.833·37-s − 1.79·38-s − 2.72·40-s − 0.971·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 - 8.35e4T \) |
| good | 2 | \( 1 - 42.0T + 512T^{2} \) |
| 5 | \( 1 + 1.40e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.13e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.14e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.15e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.05e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.50e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.75e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.15e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.02e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.95e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.34e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.45e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.71e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.75e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.33e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.87e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.41e8T + 7.60e17T^{2} \) |
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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
−11.22114170954382980494368679624, −10.21737693666967388852718093880, −8.332981505405896948352301654476, −7.17450939460115925320039196969, −6.24815817295025247075587654112, −5.18793898897200799460619493669, −3.90033699984041824433454194046, −3.42124435144762279260330370078, −2.03064062945359992675557714565, 0,
2.03064062945359992675557714565, 3.42124435144762279260330370078, 3.90033699984041824433454194046, 5.18793898897200799460619493669, 6.24815817295025247075587654112, 7.17450939460115925320039196969, 8.332981505405896948352301654476, 10.21737693666967388852718093880, 11.22114170954382980494368679624
