| L(s) = 1 | − 2i·2-s − 3·3-s − 4·4-s − 16i·5-s + 6i·6-s − 28i·7-s + 8i·8-s + 9·9-s − 32·10-s − 34i·11-s + 12·12-s − 56·14-s + 48i·15-s + 16·16-s − 138·17-s − 18i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 1.43i·5-s + 0.408i·6-s − 1.51i·7-s + 0.353i·8-s + 0.333·9-s − 1.01·10-s − 0.931i·11-s + 0.288·12-s − 1.06·14-s + 0.826i·15-s + 0.250·16-s − 1.96·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5450006466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5450006466\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 16iT - 125T^{2} \) |
| 7 | \( 1 + 28iT - 343T^{2} \) |
| 11 | \( 1 + 34iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 138T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 52T + 1.21e4T^{2} \) |
| 29 | \( 1 + 190T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 342iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 240iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 140T + 7.95e4T^{2} \) |
| 47 | \( 1 + 454iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 198T + 1.48e5T^{2} \) |
| 59 | \( 1 - 154iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 34T + 2.26e5T^{2} \) |
| 67 | \( 1 + 656iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 550iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 614iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 762iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 444iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3iT - 9.12e5T^{2} \) |
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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
−9.011585487232100486560536652263, −8.211367987608459306563103732811, −7.28503268234780715067396315840, −6.10880223402085791293615999793, −5.19526938724055731517682912111, −4.22884134726092273298243111695, −3.81105691508815134040007928694, −1.91891090373420504086766415370, −0.839830219319857879789610829901, −0.19050049684733036992718509763,
2.09379023941638925179108878765, 2.92803456477323796842886433628, 4.43737969058954829050294369255, 5.24507219402072125025670944275, 6.26419200367621389174183168414, 6.79481003860633835414604655134, 7.39848602089859828473711804103, 8.766814320113552437371812926660, 9.244353212477089123287097311358, 10.29058952452947350731258830112
