| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s + 2·13-s − 14-s + 16-s + 3·17-s + 18-s + 2·19-s − 21-s − 23-s + 24-s + 2·26-s + 27-s − 28-s + 9·29-s − 10·31-s + 32-s + 3·34-s + 36-s + 11·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s − 1.79·31-s + 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.80·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.886406483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.886406483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.349300711835563670740425205988, −7.937156705554867691859168622097, −6.99535482217755614986865443256, −6.34516048850235012007737676278, −5.54429847923358013611830820295, −4.70586181675141721672120670621, −3.76832686949922865967787924483, −3.19556541482137519648986671948, −2.27712057566840263929378719324, −1.08409241778139482879615046752,
1.08409241778139482879615046752, 2.27712057566840263929378719324, 3.19556541482137519648986671948, 3.76832686949922865967787924483, 4.70586181675141721672120670621, 5.54429847923358013611830820295, 6.34516048850235012007737676278, 6.99535482217755614986865443256, 7.937156705554867691859168622097, 8.349300711835563670740425205988
