L(s) = 1 | + 2.44·2-s + 3-s + 3.99·4-s − 5-s + 2.44·6-s − 7-s + 4.89·8-s + 9-s − 2.44·10-s − 2.44·11-s + 3.99·12-s − 0.449·13-s − 2.44·14-s − 15-s + 3.99·16-s − 0.550·17-s + 2.44·18-s − 0.449·19-s − 3.99·20-s − 21-s − 5.99·22-s − 23-s + 4.89·24-s + 25-s − 1.10·26-s + 27-s − 3.99·28-s + ⋯ |
L(s) = 1 | + 1.73·2-s + 0.577·3-s + 1.99·4-s − 0.447·5-s + 0.999·6-s − 0.377·7-s + 1.73·8-s + 0.333·9-s − 0.774·10-s − 0.738·11-s + 1.15·12-s − 0.124·13-s − 0.654·14-s − 0.258·15-s + 0.999·16-s − 0.133·17-s + 0.577·18-s − 0.103·19-s − 0.894·20-s − 0.218·21-s − 1.27·22-s − 0.208·23-s + 0.999·24-s + 0.200·25-s − 0.215·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.510051679\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.510051679\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 + 0.550T + 17T^{2} \) |
| 19 | \( 1 + 0.449T + 19T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 9.89T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 0.550T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3.55T + 47T^{2} \) |
| 53 | \( 1 - 5.44T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 9.24T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−11.86077193248076654356303449390, −10.90408889797468358163310340330, −9.878449378802652331564137511855, −8.485132080751759459239239063870, −7.43318769085195749027037335195, −6.51786511086534937764587802081, −5.37555937173621067054577784413, −4.35770431936673725435082309572, −3.38598750525546634169084841382, −2.38821824858040218890160403827,
2.38821824858040218890160403827, 3.38598750525546634169084841382, 4.35770431936673725435082309572, 5.37555937173621067054577784413, 6.51786511086534937764587802081, 7.43318769085195749027037335195, 8.485132080751759459239239063870, 9.878449378802652331564137511855, 10.90408889797468358163310340330, 11.86077193248076654356303449390