| L(s) = 1 | − 0.414·3-s − 0.585·5-s − 0.585·7-s − 2.82·9-s + 4.24·11-s − 3.82·13-s + 0.242·15-s − 2.58·17-s − 0.828·19-s + 0.242·21-s − 23-s − 4.65·25-s + 2.41·27-s − 5·29-s − 9.24·31-s − 1.75·33-s + 0.343·35-s + 5.07·37-s + 1.58·39-s − 1.34·41-s + 4·43-s + 1.65·45-s − 5.24·47-s − 6.65·49-s + 1.07·51-s − 9.31·53-s − 2.48·55-s + ⋯ |
| L(s) = 1 | − 0.239·3-s − 0.261·5-s − 0.221·7-s − 0.942·9-s + 1.27·11-s − 1.06·13-s + 0.0626·15-s − 0.627·17-s − 0.190·19-s + 0.0529·21-s − 0.208·23-s − 0.931·25-s + 0.464·27-s − 0.928·29-s − 1.66·31-s − 0.305·33-s + 0.0580·35-s + 0.833·37-s + 0.253·39-s − 0.209·41-s + 0.609·43-s + 0.246·45-s − 0.764·47-s − 0.950·49-s + 0.149·51-s − 1.27·53-s − 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 - 7.82T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−9.740226917698091581898934102727, −9.244696166204226737522278002078, −8.253481422507343259968313360094, −7.28422661601670115727527676508, −6.39332203470069533872221703447, −5.53759881019906287542947452856, −4.39067481870069225882672765725, −3.38957771621107220091477085205, −2.01585620581929199905505545309, 0,
2.01585620581929199905505545309, 3.38957771621107220091477085205, 4.39067481870069225882672765725, 5.53759881019906287542947452856, 6.39332203470069533872221703447, 7.28422661601670115727527676508, 8.253481422507343259968313360094, 9.244696166204226737522278002078, 9.740226917698091581898934102727
