L(s) = 1 | − 2.41·2-s − 3-s + 3.84·4-s + 2.41·6-s − 1.22·7-s − 4.44·8-s + 9-s + 1.58·11-s − 3.84·12-s − 3.25·13-s + 2.96·14-s + 3.06·16-s + 0.423·17-s − 2.41·18-s − 3.03·19-s + 1.22·21-s − 3.82·22-s + 1.82·23-s + 4.44·24-s + 7.87·26-s − 27-s − 4.71·28-s − 1.84·29-s + 31-s + 1.48·32-s − 1.58·33-s − 1.02·34-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.92·4-s + 0.986·6-s − 0.463·7-s − 1.57·8-s + 0.333·9-s + 0.476·11-s − 1.10·12-s − 0.903·13-s + 0.792·14-s + 0.766·16-s + 0.102·17-s − 0.569·18-s − 0.696·19-s + 0.267·21-s − 0.815·22-s + 0.381·23-s + 0.907·24-s + 1.54·26-s − 0.192·27-s − 0.890·28-s − 0.342·29-s + 0.179·31-s + 0.261·32-s − 0.275·33-s − 0.175·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4227097401\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4227097401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 0.423T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 7.30T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 + 0.633T + 47T^{2} \) |
| 53 | \( 1 - 7.83T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 - 0.133T + 61T^{2} \) |
| 67 | \( 1 - 1.68T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 - 3.49T + 79T^{2} \) |
| 83 | \( 1 - 6.68T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−9.162516800856145581624094564080, −8.314464545978093108629179000198, −7.53847261739871873653511296725, −6.83862227560549295130633619144, −6.30204389072211920339576155938, −5.23693423628893011046864221885, −4.11958288412277251972732306177, −2.78922545652671571735868087398, −1.76219958470532393673548961592, −0.55268525254077600445368116532,
0.55268525254077600445368116532, 1.76219958470532393673548961592, 2.78922545652671571735868087398, 4.11958288412277251972732306177, 5.23693423628893011046864221885, 6.30204389072211920339576155938, 6.83862227560549295130633619144, 7.53847261739871873653511296725, 8.314464545978093108629179000198, 9.162516800856145581624094564080