| L(s) = 1 | − 2.47·3-s − 1.47·5-s + 2.64·7-s + 3.11·9-s + 6.34·11-s + 5.34·13-s + 3.64·15-s − 0.715·17-s − 3.28·19-s − 6.53·21-s + 0.885·23-s − 2.83·25-s − 0.284·27-s + 0.885·29-s + 7.47·31-s − 15.6·33-s − 3.89·35-s + 37-s − 13.2·39-s − 7.75·41-s − 10.1·43-s − 4.58·45-s + 11.8·47-s − 0.0194·49-s + 1.77·51-s − 4.15·53-s − 9.34·55-s + ⋯ |
| L(s) = 1 | − 1.42·3-s − 0.658·5-s + 0.998·7-s + 1.03·9-s + 1.91·11-s + 1.48·13-s + 0.940·15-s − 0.173·17-s − 0.753·19-s − 1.42·21-s + 0.184·23-s − 0.566·25-s − 0.0546·27-s + 0.164·29-s + 1.34·31-s − 2.73·33-s − 0.657·35-s + 0.164·37-s − 2.11·39-s − 1.21·41-s − 1.54·43-s − 0.683·45-s + 1.72·47-s − 0.00277·49-s + 0.247·51-s − 0.570·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255426323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255426323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 0.715T + 17T^{2} \) |
| 19 | \( 1 + 3.28T + 19T^{2} \) |
| 23 | \( 1 - 0.885T + 23T^{2} \) |
| 29 | \( 1 - 0.885T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.637859036832325729366260988326, −8.463378371735658389147601301568, −7.23117796287311182617567450480, −6.41056871307618984602622840584, −6.07911294333867206062948173002, −4.94714307768747600552402833158, −4.27231575117588978167928527891, −3.59980817517105006491822333411, −1.68710403698863622544783417789, −0.845578287138297305164697677004,
0.845578287138297305164697677004, 1.68710403698863622544783417789, 3.59980817517105006491822333411, 4.27231575117588978167928527891, 4.94714307768747600552402833158, 6.07911294333867206062948173002, 6.41056871307618984602622840584, 7.23117796287311182617567450480, 8.463378371735658389147601301568, 8.637859036832325729366260988326
