L(s) = 1 | + 0.835·3-s + 0.221·5-s − 4.69·7-s − 2.30·9-s + 11-s + 5.89·13-s + 0.185·15-s + 7.06·17-s − 19-s − 3.92·21-s − 1.06·23-s − 4.95·25-s − 4.42·27-s − 7.62·29-s − 0.901·31-s + 0.835·33-s − 1.04·35-s − 2.71·37-s + 4.91·39-s + 0.788·41-s − 0.714·43-s − 0.511·45-s − 3.96·47-s + 15.0·49-s + 5.90·51-s − 9.69·53-s + 0.221·55-s + ⋯ |
L(s) = 1 | + 0.482·3-s + 0.0992·5-s − 1.77·7-s − 0.767·9-s + 0.301·11-s + 1.63·13-s + 0.0478·15-s + 1.71·17-s − 0.229·19-s − 0.856·21-s − 0.221·23-s − 0.990·25-s − 0.852·27-s − 1.41·29-s − 0.161·31-s + 0.145·33-s − 0.176·35-s − 0.446·37-s + 0.787·39-s + 0.123·41-s − 0.108·43-s − 0.0761·45-s − 0.578·47-s + 2.15·49-s + 0.826·51-s − 1.33·53-s + 0.0299·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 0.835T + 3T^{2} \) |
| 5 | \( 1 - 0.221T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 7.06T + 17T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 7.62T + 29T^{2} \) |
| 31 | \( 1 + 0.901T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 - 0.788T + 41T^{2} \) |
| 43 | \( 1 + 0.714T + 43T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 - 3.13T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 8.14T + 97T^{2} \) |
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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
−8.371010320180607312036264360503, −7.56732757565399523899053686655, −6.66987062583104070161668877337, −5.80789049061166002700036477353, −5.70397878422069192105914951361, −3.86251741941895803089638472868, −3.53912105631211842343381179690, −2.81686904548712357082175024326, −1.48995287513377870948440849954, 0,
1.48995287513377870948440849954, 2.81686904548712357082175024326, 3.53912105631211842343381179690, 3.86251741941895803089638472868, 5.70397878422069192105914951361, 5.80789049061166002700036477353, 6.66987062583104070161668877337, 7.56732757565399523899053686655, 8.371010320180607312036264360503