| L(s) = 1 | + 3.41·5-s + 0.828·11-s + 4.24·13-s + 7.41·17-s + 6.82·19-s − 4.82·23-s + 6.65·25-s − 2.82·29-s − 2.82·31-s − 1.65·37-s + 10.2·41-s + 11.3·43-s + 4.48·47-s + 2·53-s + 2.82·55-s − 8.48·59-s − 11.0·61-s + 14.4·65-s − 11.3·67-s − 10.4·71-s − 7.75·73-s − 13.6·79-s + 4·83-s + 25.3·85-s + 5.75·89-s + 23.3·95-s + 0.242·97-s + ⋯ |
| L(s) = 1 | + 1.52·5-s + 0.249·11-s + 1.17·13-s + 1.79·17-s + 1.56·19-s − 1.00·23-s + 1.33·25-s − 0.525·29-s − 0.508·31-s − 0.272·37-s + 1.59·41-s + 1.72·43-s + 0.654·47-s + 0.274·53-s + 0.381·55-s − 1.10·59-s − 1.41·61-s + 1.79·65-s − 1.38·67-s − 1.24·71-s − 0.907·73-s − 1.53·79-s + 0.439·83-s + 2.74·85-s + 0.610·89-s + 2.39·95-s + 0.0246·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.533171331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.533171331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.75T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 - 0.242T + 97T^{2} \) |
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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
−7.61521599329506929466557052016, −7.52875208988564582036711018828, −6.18126367119625366455855257133, −5.81932365823782292937544258576, −5.50798605144875358760445915357, −4.34819313248293558788317763399, −3.44744830979579705900893433965, −2.74011801101535437561929336234, −1.60582103892732957435416612135, −1.09649935208597408983649326140,
1.09649935208597408983649326140, 1.60582103892732957435416612135, 2.74011801101535437561929336234, 3.44744830979579705900893433965, 4.34819313248293558788317763399, 5.50798605144875358760445915357, 5.81932365823782292937544258576, 6.18126367119625366455855257133, 7.52875208988564582036711018828, 7.61521599329506929466557052016
