| L(s) = 1 | + 2.44·5-s + 1.41·7-s − 3.46·11-s − 4.89·13-s − 4·17-s + 6.92·19-s − 5.65·23-s + 0.999·25-s + 2.44·29-s − 1.41·31-s + 3.46·35-s − 4.89·37-s − 4·41-s − 6.92·43-s + 5.65·47-s − 5·49-s − 7.34·53-s − 8.48·55-s + 13.8·59-s − 4.89·61-s − 11.9·65-s − 11.3·71-s − 4·73-s − 4.89·77-s + 7.07·79-s − 10.3·83-s − 9.79·85-s + ⋯ |
| L(s) = 1 | + 1.09·5-s + 0.534·7-s − 1.04·11-s − 1.35·13-s − 0.970·17-s + 1.58·19-s − 1.17·23-s + 0.199·25-s + 0.454·29-s − 0.254·31-s + 0.585·35-s − 0.805·37-s − 0.624·41-s − 1.05·43-s + 0.825·47-s − 0.714·49-s − 1.00·53-s − 1.14·55-s + 1.80·59-s − 0.627·61-s − 1.48·65-s − 1.34·71-s − 0.468·73-s − 0.558·77-s + 0.795·79-s − 1.14·83-s − 1.06·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.891597724874808856781112970842, −7.29544850865685846767407329048, −6.50922740917815331884845607637, −5.51672857862294356425316622623, −5.19249205268911090698096068860, −4.40932880051237344618820879619, −3.11475485408224554754741143931, −2.31894169013915965413894863647, −1.63655580842767063566560863814, 0,
1.63655580842767063566560863814, 2.31894169013915965413894863647, 3.11475485408224554754741143931, 4.40932880051237344618820879619, 5.19249205268911090698096068860, 5.51672857862294356425316622623, 6.50922740917815331884845607637, 7.29544850865685846767407329048, 7.891597724874808856781112970842
