| L(s) = 1 | − 0.414·2-s − 1.82·4-s − 5-s − 0.828·7-s + 1.58·8-s + 0.414·10-s − 0.585·11-s − 13-s + 0.343·14-s + 3·16-s + 4.82·17-s + 3.41·19-s + 1.82·20-s + 0.242·22-s + 1.41·23-s + 25-s + 0.414·26-s + 1.51·28-s − 5.65·29-s + 10.2·31-s − 4.41·32-s − 1.99·34-s + 0.828·35-s + 8.48·37-s − 1.41·38-s − 1.58·40-s + 8.82·41-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.313·7-s + 0.560·8-s + 0.130·10-s − 0.176·11-s − 0.277·13-s + 0.0917·14-s + 0.750·16-s + 1.17·17-s + 0.783·19-s + 0.408·20-s + 0.0517·22-s + 0.294·23-s + 0.200·25-s + 0.0812·26-s + 0.286·28-s − 1.05·29-s + 1.83·31-s − 0.780·32-s − 0.342·34-s + 0.140·35-s + 1.39·37-s − 0.229·38-s − 0.250·40-s + 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8877276736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8877276736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−10.49217856516854857244707613659, −9.714286165109304329331025621075, −9.094636004158736703950196942447, −7.931217069701776126282701574024, −7.52806058553861407863894042151, −6.06463888489816548881869814765, −5.06077252307295835194499274723, −4.08089846690771216958976744620, −2.97450618779523508374006928154, −0.897256041534199990641944032128,
0.897256041534199990641944032128, 2.97450618779523508374006928154, 4.08089846690771216958976744620, 5.06077252307295835194499274723, 6.06463888489816548881869814765, 7.52806058553861407863894042151, 7.931217069701776126282701574024, 9.094636004158736703950196942447, 9.714286165109304329331025621075, 10.49217856516854857244707613659
