| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s − 8·8-s + 9·9-s − 10·10-s − 12·12-s − 26·13-s − 15·15-s + 16·16-s − 18·17-s − 18·18-s − 92·19-s + 20·20-s + 24·24-s + 25·25-s + 52·26-s − 27·27-s − 6·29-s + 30·30-s + 4·31-s − 32·32-s + 36·34-s + 36·36-s + 410·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.256·17-s − 0.235·18-s − 1.11·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.0384·29-s + 0.182·30-s + 0.0231·31-s − 0.176·32-s + 0.181·34-s + 1/6·36-s + 1.82·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9945185110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9945185110\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 4 T + p^{3} T^{2} \) |
| 37 | \( 1 - 410 T + p^{3} T^{2} \) |
| 41 | \( 1 + 174 T + p^{3} T^{2} \) |
| 43 | \( 1 - 248 T + p^{3} T^{2} \) |
| 47 | \( 1 + 420 T + p^{3} T^{2} \) |
| 53 | \( 1 - 102 T + p^{3} T^{2} \) |
| 59 | \( 1 - 588 T + p^{3} T^{2} \) |
| 61 | \( 1 + 650 T + p^{3} T^{2} \) |
| 67 | \( 1 - 152 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 610 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1048 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 834 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−9.266835220076824405130197145562, −8.378821591508209289226123958622, −7.55909315232425014867749020512, −6.64725523501129945758546601190, −6.07725904430497976392902678493, −5.08538754910944017248766008144, −4.14174019396440737165115938041, −2.72364688812667090971052565263, −1.78768835837659023776048208158, −0.55499738254595109877632348708,
0.55499738254595109877632348708, 1.78768835837659023776048208158, 2.72364688812667090971052565263, 4.14174019396440737165115938041, 5.08538754910944017248766008144, 6.07725904430497976392902678493, 6.64725523501129945758546601190, 7.55909315232425014867749020512, 8.378821591508209289226123958622, 9.266835220076824405130197145562
