| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2i·7-s + i·8-s − 9-s − 0.763·11-s − i·12-s − 1.85i·13-s + 2·14-s + 16-s + 1.14i·17-s + i·18-s + 7.23·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 0.230·11-s − 0.288i·12-s − 0.514i·13-s + 0.534·14-s + 0.250·16-s + 0.277i·17-s + 0.235i·18-s + 1.66·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676154841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676154841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 1.85iT - 13T^{2} \) |
| 17 | \( 1 - 1.14iT - 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 8.85iT - 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 3.23iT - 43T^{2} \) |
| 47 | \( 1 - 9.23iT - 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 - 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 9.85iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 + 7.14iT - 97T^{2} \) |
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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
−8.846901121000258444700481579934, −7.915391542574893066370679683165, −7.20075721999886133659022516655, −5.87025671298525871327932164869, −5.51475481878090502954690199630, −4.63877189670983758577076197219, −3.77096261010516329178413532741, −2.93167212449551053568073219845, −2.23496479814391807421934615549, −0.815296902955333526087965922843,
0.71628230225356980410554219470, 1.77304568504698468588190871500, 3.16654381156014955891190649040, 3.86669965986909404876031540522, 5.01852125174198936136235690463, 5.48657873112552013723314203024, 6.49988627880221397461661548440, 7.15012782867225758507818635910, 7.57276066427401505813351763241, 8.265708170423666656098834452037
