L(s) = 1 | + (0.846 − 1.75i)5-s + (−0.974 + 0.222i)9-s + (1.21 + 0.277i)13-s + (−0.467 − 0.467i)17-s + (−1.74 − 2.19i)25-s + (−0.974 − 0.222i)29-s + (1.68 + 1.05i)37-s + (−0.158 + 0.158i)41-s + (−0.433 + 1.90i)45-s + (0.222 + 0.974i)49-s + (−0.781 − 0.376i)53-s + (1.87 − 0.211i)61-s + (1.51 − 1.90i)65-s + (−0.623 + 1.78i)73-s + (0.900 − 0.433i)81-s + ⋯ |
L(s) = 1 | + (0.846 − 1.75i)5-s + (−0.974 + 0.222i)9-s + (1.21 + 0.277i)13-s + (−0.467 − 0.467i)17-s + (−1.74 − 2.19i)25-s + (−0.974 − 0.222i)29-s + (1.68 + 1.05i)37-s + (−0.158 + 0.158i)41-s + (−0.433 + 1.90i)45-s + (0.222 + 0.974i)49-s + (−0.781 − 0.376i)53-s + (1.87 − 0.211i)61-s + (1.51 − 1.90i)65-s + (−0.623 + 1.78i)73-s + (0.900 − 0.433i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.091705210\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.091705210\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (0.974 + 0.222i)T \) |
good | 3 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 5 | \( 1 + (-0.846 + 1.75i)T + (-0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 13 | \( 1 + (-1.21 - 0.277i)T + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (0.467 + 0.467i)T + iT^{2} \) |
| 19 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 23 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 37 | \( 1 + (-1.68 - 1.05i)T + (0.433 + 0.900i)T^{2} \) |
| 41 | \( 1 + (0.158 - 0.158i)T - iT^{2} \) |
| 43 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 47 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 53 | \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1.87 + 0.211i)T + (0.974 - 0.222i)T^{2} \) |
| 67 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.623 - 1.78i)T + (-0.781 - 0.623i)T^{2} \) |
| 79 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.559 - 1.59i)T + (-0.781 + 0.623i)T^{2} \) |
| 97 | \( 1 + (1.97 + 0.222i)T + (0.974 + 0.222i)T^{2} \) |
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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
−9.846781758125136840473988378341, −9.216984811974811928240770285218, −8.560806845031130285271335805028, −7.979896886397177014378245475234, −6.41031375069102620664831790076, −5.72124758511507812802887384740, −4.97516847316289865906735086964, −4.01334883377008655683312628203, −2.44129805991302587478381299853, −1.18672375782254898786166719241,
2.03302391219050904038256123295, 3.02535038996027162754414776790, 3.82786386909435779796869105778, 5.63442687226350527913778288997, 6.07864482460356253272143398860, 6.84165730739571425838368140689, 7.82394688944776467652884424874, 8.850622856478770288199633279811, 9.648565295487022330584666136006, 10.61434751136575621531592638984