| L(s) = 1 | − 2.96·5-s + 4.15·7-s − 3.19·11-s − 5.35·13-s − 17-s + 1.61·19-s + 8.46·23-s + 3.77·25-s + 6.96·29-s + 6.54·31-s − 12.3·35-s − 6.96·37-s − 8.70·41-s + 4.31·43-s − 5.92·47-s + 10.2·49-s − 4.70·53-s + 9.46·55-s + 7.53·59-s − 10.1·61-s + 15.8·65-s − 3.22·67-s − 13.7·71-s + 5.22·73-s − 13.2·77-s − 13.2·79-s + 4.31·83-s + ⋯ |
| L(s) = 1 | − 1.32·5-s + 1.57·7-s − 0.963·11-s − 1.48·13-s − 0.242·17-s + 0.369·19-s + 1.76·23-s + 0.755·25-s + 1.29·29-s + 1.17·31-s − 2.08·35-s − 1.14·37-s − 1.35·41-s + 0.657·43-s − 0.864·47-s + 1.46·49-s − 0.645·53-s + 1.27·55-s + 0.981·59-s − 1.30·61-s + 1.96·65-s − 0.393·67-s − 1.63·71-s + 0.611·73-s − 1.51·77-s − 1.49·79-s + 0.473·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 + 3.19T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 29 | \( 1 - 6.96T + 29T^{2} \) |
| 31 | \( 1 - 6.54T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 - 4.31T + 43T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 5.22T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + 2.77T + 97T^{2} \) |
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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
−7.941809484926371612945198438187, −7.34367487338673502671339679942, −6.78346169909090632775112850447, −5.33481479788559907253714184498, −4.79191300904464210334878864030, −4.51863174312756331143547324655, −3.21730293777465648369628975693, −2.52943585135309431949769048058, −1.27939297748265963050871482163, 0,
1.27939297748265963050871482163, 2.52943585135309431949769048058, 3.21730293777465648369628975693, 4.51863174312756331143547324655, 4.79191300904464210334878864030, 5.33481479788559907253714184498, 6.78346169909090632775112850447, 7.34367487338673502671339679942, 7.941809484926371612945198438187
