| L(s) = 1 | − 1.61i·2-s + i·3-s − 0.618·4-s + 1.61·6-s + 2i·7-s − 2.23i·8-s − 9-s − 3·11-s − 0.618i·12-s − i·13-s + 3.23·14-s − 4.85·16-s + 4.23i·17-s + 1.61i·18-s + 6.70·19-s + ⋯ |
| L(s) = 1 | − 1.14i·2-s + 0.577i·3-s − 0.309·4-s + 0.660·6-s + 0.755i·7-s − 0.790i·8-s − 0.333·9-s − 0.904·11-s − 0.178i·12-s − 0.277i·13-s + 0.864·14-s − 1.21·16-s + 1.02i·17-s + 0.381i·18-s + 1.53·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664403886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664403886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.61iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 4.23iT - 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 5.38iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 7.38iT - 43T^{2} \) |
| 47 | \( 1 - 4.76iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 9.14T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 3.85iT - 97T^{2} \) |
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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
−9.577266756812391563194571817788, −8.634806059131835752279961542190, −7.896405783503277378885058156493, −6.83861716956208763432138480728, −5.74569689445199836623490729166, −5.11772116988684030709242722812, −3.98773615507736036836740811364, −3.11247991479339906016379191630, −2.48214053294405845939642355522, −1.19020297183166260820253643671,
0.67312396664317958234246405237, 2.27319523376607697393281736405, 3.23772062003150715193663704871, 4.82297160402479758201964185540, 5.17062265146406838790021623230, 6.43944918541099985455741588931, 6.76563599382677668845501300381, 7.71655357321589657165967726719, 7.968598638277173610037773116367, 8.936613180115775222643493144080
