L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 2.14i·11-s + (−2.16 + 2.16i)13-s + 1.00·14-s − 1.00·16-s + (4.46 − 4.46i)17-s + (1.51 + 1.51i)22-s + (6.32 + 6.32i)23-s + 3.05i·26-s + (0.707 − 0.707i)28-s − 8.20·29-s − 1.85·31-s + (−0.707 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.647i·11-s + (−0.599 + 0.599i)13-s + 0.267·14-s − 0.250·16-s + (1.08 − 1.08i)17-s + (0.323 + 0.323i)22-s + (1.31 + 1.31i)23-s + 0.599i·26-s + (0.133 − 0.133i)28-s − 1.52·29-s − 0.332·31-s + (−0.125 + 0.125i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.399530177\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.399530177\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 2.14iT - 11T^{2} \) |
| 13 | \( 1 + (2.16 - 2.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.46 + 4.46i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-6.32 - 6.32i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.20T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 + (-5.13 - 5.13i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.56iT - 41T^{2} \) |
| 43 | \( 1 + (-7.84 + 7.84i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.01 - 5.01i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.35 - 2.35i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 + (-3.84 - 3.84i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.420iT - 71T^{2} \) |
| 73 | \( 1 + (-2.63 + 2.63i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.23iT - 79T^{2} \) |
| 83 | \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + (-0.606 - 0.606i)T + 97iT^{2} \) |
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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
−9.013718315815437440825702168393, −7.62109288021923122574017038466, −7.37723799716314148329228558171, −6.33495017672180265565162812674, −5.28581808829466715393522522990, −4.99108625820172357406749940440, −3.94326685021646214639370664395, −3.05337430997156636288091371999, −2.16518997249384665978596856208, −1.12520030140201723699817456950,
0.71406542007926834295524987654, 2.20954131877944094732523088243, 3.28438097186458471600921194615, 3.96309782515593077645476213269, 4.97139580081825624594661180254, 5.61705771207942796330616677631, 6.28980480200662486608576300590, 7.28059514408427202296642704864, 7.76910578921765018474912773564, 8.519235454547818037394477700795