| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 13-s + 14-s + 16-s + 3·17-s + 2·19-s + 3·23-s − 26-s + 28-s + 3·29-s − 31-s + 32-s + 3·34-s + 2·37-s + 2·38-s − 3·41-s − 7·43-s + 3·46-s + 6·47-s + 49-s − 52-s + 9·53-s + 56-s + 3·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.625·23-s − 0.196·26-s + 0.188·28-s + 0.557·29-s − 0.179·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s + 0.324·38-s − 0.468·41-s − 1.06·43-s + 0.442·46-s + 0.875·47-s + 1/7·49-s − 0.138·52-s + 1.23·53-s + 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.265298908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.265298908\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.574617760662529959095220674736, −7.81105115349060706740571682353, −7.13287375202220998208663000365, −6.37921532248785260059826190315, −5.41672807926563197816250352466, −4.96706816419789722014449523692, −3.97978659211673864480007661391, −3.16555161271588275796959360507, −2.22231489114057328774149093907, −1.03374611869035395577761520026,
1.03374611869035395577761520026, 2.22231489114057328774149093907, 3.16555161271588275796959360507, 3.97978659211673864480007661391, 4.96706816419789722014449523692, 5.41672807926563197816250352466, 6.37921532248785260059826190315, 7.13287375202220998208663000365, 7.81105115349060706740571682353, 8.574617760662529959095220674736
