| L(s) = 1 | − 2.41·3-s + 1.41·5-s + 2.82·9-s + 11-s + 3.82·13-s − 3.41·15-s + 2·17-s + 5.41·19-s + 2.58·23-s − 2.99·25-s + 0.414·27-s + 29-s − 1.65·31-s − 2.41·33-s + 5.07·37-s − 9.24·39-s + 2.24·41-s − 8·43-s + 4·45-s + 0.828·47-s − 4.82·51-s + 5.41·53-s + 1.41·55-s − 13.0·57-s − 1.58·59-s + 13.8·61-s + 5.41·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.632·5-s + 0.942·9-s + 0.301·11-s + 1.06·13-s − 0.881·15-s + 0.485·17-s + 1.24·19-s + 0.539·23-s − 0.599·25-s + 0.0797·27-s + 0.185·29-s − 0.297·31-s − 0.420·33-s + 0.833·37-s − 1.48·39-s + 0.350·41-s − 1.21·43-s + 0.596·45-s + 0.120·47-s − 0.676·51-s + 0.743·53-s + 0.190·55-s − 1.73·57-s − 0.206·59-s + 1.77·61-s + 0.671·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481517320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481517320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 - 5.41T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 6.07T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 0.585T + 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 + 6.48T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.408613426314444632427157178316, −7.44827188279226687506810424334, −6.73765598686940901866980966849, −5.96714205034201947471732724292, −5.63272589180528619529886886552, −4.89696705184936992551818598548, −3.92876176170403645168625307302, −2.97366214652207885992025212658, −1.59227931259082451993751163589, −0.798526390287768349675557604642,
0.798526390287768349675557604642, 1.59227931259082451993751163589, 2.97366214652207885992025212658, 3.92876176170403645168625307302, 4.89696705184936992551818598548, 5.63272589180528619529886886552, 5.96714205034201947471732724292, 6.73765598686940901866980966849, 7.44827188279226687506810424334, 8.408613426314444632427157178316
