L(s) = 1 | − 0.209·2-s − 1.61·3-s − 0.956·4-s + 1.82·5-s + 0.338·6-s + 0.408·8-s + 1.61·9-s − 0.381·10-s + 1.33·11-s + 1.54·12-s − 2.95·15-s + 0.870·16-s − 17-s − 0.338·18-s − 1.74·20-s − 0.279·22-s − 0.661·24-s + 2.33·25-s − 27-s − 1.95·29-s + 0.618·30-s − 0.209·31-s − 0.591·32-s − 2.16·33-s + 0.209·34-s − 1.54·36-s + 0.747·40-s + ⋯ |
L(s) = 1 | − 0.209·2-s − 1.61·3-s − 0.956·4-s + 1.82·5-s + 0.338·6-s + 0.408·8-s + 1.61·9-s − 0.381·10-s + 1.33·11-s + 1.54·12-s − 2.95·15-s + 0.870·16-s − 17-s − 0.338·18-s − 1.74·20-s − 0.279·22-s − 0.661·24-s + 2.33·25-s − 27-s − 1.95·29-s + 0.618·30-s − 0.209·31-s − 0.591·32-s − 2.16·33-s + 0.209·34-s − 1.54·36-s + 0.747·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4769372595\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4769372595\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 239 | \( 1 - T \) |
good | 2 | \( 1 + 0.209T + T^{2} \) |
| 3 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - 1.82T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.33T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.95T + T^{2} \) |
| 31 | \( 1 + 0.209T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.95T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−12.44482906154760982008960274212, −11.25765959295992078862197464274, −10.43149203132747921694537354646, −9.511910133459389071755600150432, −8.981340611610185972617487201368, −6.96859303265943508864605154875, −6.03386551036409625280310185267, −5.39468122066492703540558794955, −4.27753201481372775171416648494, −1.54432900565373525411282010947,
1.54432900565373525411282010947, 4.27753201481372775171416648494, 5.39468122066492703540558794955, 6.03386551036409625280310185267, 6.96859303265943508864605154875, 8.981340611610185972617487201368, 9.511910133459389071755600150432, 10.43149203132747921694537354646, 11.25765959295992078862197464274, 12.44482906154760982008960274212