L(s) = 1 | − 3.35·5-s − 2.96·7-s − 1.61·11-s + 2·13-s − 4.96·17-s − 1.35·19-s + 23-s + 6.22·25-s + 7.92·29-s + 5.92·31-s + 9.92·35-s − 2.31·37-s + 1.22·41-s − 4.57·43-s − 1.92·47-s + 1.77·49-s + 4.12·53-s + 5.40·55-s − 2.70·59-s + 14.3·61-s − 6.70·65-s + 4.57·67-s + 16.6·71-s + 2·73-s + 4.77·77-s − 5.03·79-s + 15.0·83-s + ⋯ |
L(s) = 1 | − 1.49·5-s − 1.11·7-s − 0.486·11-s + 0.554·13-s − 1.20·17-s − 0.309·19-s + 0.208·23-s + 1.24·25-s + 1.47·29-s + 1.06·31-s + 1.67·35-s − 0.380·37-s + 0.191·41-s − 0.697·43-s − 0.280·47-s + 0.253·49-s + 0.566·53-s + 0.728·55-s − 0.351·59-s + 1.83·61-s − 0.831·65-s + 0.558·67-s + 1.97·71-s + 0.234·73-s + 0.544·77-s − 0.566·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7697082689\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7697082689\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3.35T + 5T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.270875022616642954554970282271, −8.407127600607740410107958171594, −7.973494313291267645424234431169, −6.71302597228446650949557981413, −6.57501418054627492575418375420, −5.11370103577305735513057210660, −4.20830482515485365725303615097, −3.47785657698841285526821800752, −2.55986016065240512488786061072, −0.58374185399782108741194283634,
0.58374185399782108741194283634, 2.55986016065240512488786061072, 3.47785657698841285526821800752, 4.20830482515485365725303615097, 5.11370103577305735513057210660, 6.57501418054627492575418375420, 6.71302597228446650949557981413, 7.973494313291267645424234431169, 8.407127600607740410107958171594, 9.270875022616642954554970282271