| L(s) = 1 | − 3·3-s + 8·5-s + 9·9-s − 40·11-s + 4·13-s − 24·15-s − 84·17-s − 148·19-s − 84·23-s − 61·25-s − 27·27-s + 58·29-s + 136·31-s + 120·33-s − 222·37-s − 12·39-s + 420·41-s + 164·43-s + 72·45-s − 488·47-s + 252·51-s + 478·53-s − 320·55-s + 444·57-s − 548·59-s + 692·61-s + 32·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.715·5-s + 1/3·9-s − 1.09·11-s + 0.0853·13-s − 0.413·15-s − 1.19·17-s − 1.78·19-s − 0.761·23-s − 0.487·25-s − 0.192·27-s + 0.371·29-s + 0.787·31-s + 0.633·33-s − 0.986·37-s − 0.0492·39-s + 1.59·41-s + 0.581·43-s + 0.238·45-s − 1.51·47-s + 0.691·51-s + 1.23·53-s − 0.784·55-s + 1.03·57-s − 1.20·59-s + 1.45·61-s + 0.0610·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.113203597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113203597\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 420 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 488 T + p^{3} T^{2} \) |
| 53 | \( 1 - 478 T + p^{3} T^{2} \) |
| 59 | \( 1 + 548 T + p^{3} T^{2} \) |
| 61 | \( 1 - 692 T + p^{3} T^{2} \) |
| 67 | \( 1 - 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 524 T + p^{3} T^{2} \) |
| 73 | \( 1 - 440 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1216 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 604 T + p^{3} T^{2} \) |
| 97 | \( 1 + 832 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.550749785123397935222612619102, −7.978669492407616052785633568171, −6.84570637421556634816037786666, −6.28186470000616105363592685365, −5.57898689299491944710236581520, −4.71800656763016009583511414981, −3.97331621931523526414903519802, −2.49515494339714906245748248791, −1.95965979515621353051533111752, −0.45931418965791266150935505808,
0.45931418965791266150935505808, 1.95965979515621353051533111752, 2.49515494339714906245748248791, 3.97331621931523526414903519802, 4.71800656763016009583511414981, 5.57898689299491944710236581520, 6.28186470000616105363592685365, 6.84570637421556634816037786666, 7.978669492407616052785633568171, 8.550749785123397935222612619102
