L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s − 13-s − 2·14-s − 15-s + 16-s + 8·17-s − 2·18-s + 20-s + 2·21-s − 3·22-s + 4·23-s − 24-s − 4·25-s − 26-s + 5·27-s − 2·28-s − 29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.471·18-s + 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.962·27-s − 0.377·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.033207509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033207509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.09322040009074974335092601729, −13.99719858770620348924737003173, −12.92256133309152763022923630908, −12.00978753036483029980569259298, −10.74800585012115946663581328742, −9.633495492381883994739439375568, −7.71436185563727090632996745083, −6.11658999858863662469236119804, −5.25136790623537531107690201816, −3.07071881135695746551942332895,
3.07071881135695746551942332895, 5.25136790623537531107690201816, 6.11658999858863662469236119804, 7.71436185563727090632996745083, 9.633495492381883994739439375568, 10.74800585012115946663581328742, 12.00978753036483029980569259298, 12.92256133309152763022923630908, 13.99719858770620348924737003173, 15.09322040009074974335092601729