L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 6·11-s + 2·13-s + 2·15-s + 6·17-s − 19-s − 4·21-s + 6·23-s − 25-s − 27-s − 4·29-s − 6·33-s − 8·35-s + 6·37-s − 2·39-s + 12·41-s − 2·45-s − 2·47-s + 9·49-s − 6·51-s − 12·53-s − 12·55-s + 57-s + 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s − 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.04·33-s − 1.35·35-s + 0.986·37-s − 0.320·39-s + 1.87·41-s − 0.298·45-s − 0.291·47-s + 9/7·49-s − 0.840·51-s − 1.64·53-s − 1.61·55-s + 0.132·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.033584150\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.033584150\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.501994221085784506502817541541, −7.63870668650087109482054051645, −7.30477523417000272051173704467, −6.19826130044277810938742089352, −5.59150094429109137122777969674, −4.52452987582281542071008706793, −4.15217436255771555307261745962, −3.21637007880483542993719877178, −1.57910401514899058156295584774, −0.994467879463985454107690875367,
0.994467879463985454107690875367, 1.57910401514899058156295584774, 3.21637007880483542993719877178, 4.15217436255771555307261745962, 4.52452987582281542071008706793, 5.59150094429109137122777969674, 6.19826130044277810938742089352, 7.30477523417000272051173704467, 7.63870668650087109482054051645, 8.501994221085784506502817541541