| L(s) = 1 | − i·2-s + 3-s − 4-s − 0.267i·5-s − i·6-s + 0.732i·7-s + i·8-s + 9-s − 0.267·10-s + 4.73i·11-s − 12-s + 0.732·14-s − 0.267i·15-s + 16-s + 2.26·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.119i·5-s − 0.408i·6-s + 0.276i·7-s + 0.353i·8-s + 0.333·9-s − 0.0847·10-s + 1.42i·11-s − 0.288·12-s + 0.195·14-s − 0.0691i·15-s + 0.250·16-s + 0.550·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.902942203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902942203\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.267iT - 5T^{2} \) |
| 7 | \( 1 - 0.732iT - 7T^{2} \) |
| 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 + 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 + 8.19iT - 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 1.26iT - 71T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−10.13813940395466382935150036008, −8.981580443690831079702957291680, −8.661764541556746334670589944013, −7.41600766659694072758545971290, −6.77446366200483207911655818405, −5.21746276857318611007526588655, −4.61927499362037629232911607305, −3.40517161812166722841184429088, −2.50179221108861253382008985472, −1.34206699551082744999465520542,
0.974084463606856009848919241523, 2.85851884625716434580661568016, 3.68097718961713866078474745502, 4.83122794061019655331900479219, 5.82474506822383070728023347452, 6.66023388774610233276104910896, 7.57538245280212718477716413805, 8.285067916059943381321756943947, 8.979504973627182053573390589985, 9.773428143542378836996848263782
