| L(s) = 1 | − 2.56·2-s + 4.56·4-s − 7-s − 6.56·8-s − 11-s + 0.438·13-s + 2.56·14-s + 7.68·16-s − 1.43·17-s + 3·19-s + 2.56·22-s − 6.56·23-s − 1.12·26-s − 4.56·28-s + 5.12·29-s + 4.68·31-s − 6.56·32-s + 3.68·34-s + 10.8·37-s − 7.68·38-s − 7.68·41-s − 4.68·43-s − 4.56·44-s + 16.8·46-s − 5.43·47-s − 6·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 2.28·4-s − 0.377·7-s − 2.31·8-s − 0.301·11-s + 0.121·13-s + 0.684·14-s + 1.92·16-s − 0.348·17-s + 0.688·19-s + 0.546·22-s − 1.36·23-s − 0.220·26-s − 0.862·28-s + 0.951·29-s + 0.841·31-s − 1.15·32-s + 0.631·34-s + 1.77·37-s − 1.24·38-s − 1.20·41-s − 0.714·43-s − 0.687·44-s + 2.47·46-s − 0.793·47-s − 0.857·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 6.68T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 2.12T + 97T^{2} \) |
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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
−8.362203898016216724423125775925, −8.168870701943329140020408059164, −7.20609466003908236918619456375, −6.52833819232382999435466687619, −5.83943773071092659340013879093, −4.55273020158778746224588984591, −3.21514001192301003929350836125, −2.35297513900005454752673839357, −1.24780105277825421698532900211, 0,
1.24780105277825421698532900211, 2.35297513900005454752673839357, 3.21514001192301003929350836125, 4.55273020158778746224588984591, 5.83943773071092659340013879093, 6.52833819232382999435466687619, 7.20609466003908236918619456375, 8.168870701943329140020408059164, 8.362203898016216724423125775925
