| L(s) = 1 | − 2.01·3-s − 4.06·5-s − 4.33·7-s + 1.04·9-s − 3.72·11-s − 4.52·13-s + 8.17·15-s − 5.00·17-s + 2.88·19-s + 8.71·21-s + 3.74·23-s + 11.5·25-s + 3.92·27-s + 9.42·29-s − 6.53·31-s + 7.48·33-s + 17.6·35-s − 6.59·37-s + 9.10·39-s − 2.05·41-s + 7.35·43-s − 4.26·45-s − 2.80·47-s + 11.7·49-s + 10.0·51-s + 11.5·53-s + 15.1·55-s + ⋯ |
| L(s) = 1 | − 1.16·3-s − 1.81·5-s − 1.63·7-s + 0.349·9-s − 1.12·11-s − 1.25·13-s + 2.11·15-s − 1.21·17-s + 0.661·19-s + 1.90·21-s + 0.779·23-s + 2.30·25-s + 0.755·27-s + 1.75·29-s − 1.17·31-s + 1.30·33-s + 2.97·35-s − 1.08·37-s + 1.45·39-s − 0.321·41-s + 1.12·43-s − 0.635·45-s − 0.409·47-s + 1.67·49-s + 1.40·51-s + 1.58·53-s + 2.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 2.01T + 3T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 + 4.33T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 6.53T + 31T^{2} \) |
| 37 | \( 1 + 6.59T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 - 7.35T + 43T^{2} \) |
| 47 | \( 1 + 2.80T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
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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
−7.30250746057203341387582216751, −7.16641035370010404306862785435, −6.45310676449487197675006517831, −5.47843610148957256812846762899, −4.85749996329290306413575128611, −4.19163464344384105138333060828, −3.14505290865627274379591555553, −2.71852553385492893102236130313, −0.56861644557585869996959433509, 0,
0.56861644557585869996959433509, 2.71852553385492893102236130313, 3.14505290865627274379591555553, 4.19163464344384105138333060828, 4.85749996329290306413575128611, 5.47843610148957256812846762899, 6.45310676449487197675006517831, 7.16641035370010404306862785435, 7.30250746057203341387582216751
