| L(s) = 1 | − i·2-s + 3-s − 4-s + 4.04i·5-s − i·6-s − 0.692i·7-s + i·8-s + 9-s + 4.04·10-s + 4.85i·11-s − 12-s − 0.692·14-s + 4.04i·15-s + 16-s − 7.38·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.81i·5-s − 0.408i·6-s − 0.261i·7-s + 0.353i·8-s + 0.333·9-s + 1.28·10-s + 1.46i·11-s − 0.288·12-s − 0.184·14-s + 1.04i·15-s + 0.250·16-s − 1.79·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311680798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311680798\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.04iT - 5T^{2} \) |
| 7 | \( 1 + 0.692iT - 7T^{2} \) |
| 11 | \( 1 - 4.85iT - 11T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 + 1.78iT - 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 + 3.34T + 29T^{2} \) |
| 31 | \( 1 - 0.972iT - 31T^{2} \) |
| 37 | \( 1 + 1.28iT - 37T^{2} \) |
| 41 | \( 1 - 1.50iT - 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 7.20iT - 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 1.30iT - 59T^{2} \) |
| 61 | \( 1 + 0.396T + 61T^{2} \) |
| 67 | \( 1 - 6.05iT - 67T^{2} \) |
| 71 | \( 1 + 1.32iT - 71T^{2} \) |
| 73 | \( 1 - 7.65iT - 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 15.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.10iT - 89T^{2} \) |
| 97 | \( 1 + 8.54iT - 97T^{2} \) |
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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
−10.18309634630888290225387004116, −9.624030967868472264902246746072, −8.666359133629152854343203103407, −7.38757951516034581991644915916, −7.10649551024888338897072314165, −6.05075647708314483753185269432, −4.44576254840836318438154960731, −3.82253387853839169425188470607, −2.54203323910779082045144166718, −2.13397469367192747351334117366,
0.53068843562592457290458618166, 2.06510733659147620384261976194, 3.77955172827098274832144802424, 4.48838879773782674869707215619, 5.53731470480394207642060446762, 6.13491554949825851366793948114, 7.47198525089832125550909266730, 8.365398230448837643681388581185, 8.798267522111578953997592293996, 9.197688213805736506823808132532
