| L(s) = 1 | + 3.12·3-s − 5-s + 1.51·7-s + 6.76·9-s − 4.24·11-s + 4.15·13-s − 3.12·15-s − 3.51·17-s + 19-s + 4.73·21-s + 8.73·23-s + 25-s + 11.7·27-s + 1.45·29-s + 4.96·31-s − 13.2·33-s − 1.51·35-s + 7.60·37-s + 12.9·39-s − 9.21·41-s + 8.31·43-s − 6.76·45-s − 5.28·47-s − 4.70·49-s − 10.9·51-s + 0.155·53-s + 4.24·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 0.447·5-s + 0.572·7-s + 2.25·9-s − 1.28·11-s + 1.15·13-s − 0.806·15-s − 0.852·17-s + 0.229·19-s + 1.03·21-s + 1.82·23-s + 0.200·25-s + 2.26·27-s + 0.270·29-s + 0.892·31-s − 2.31·33-s − 0.256·35-s + 1.25·37-s + 2.07·39-s − 1.43·41-s + 1.26·43-s − 1.00·45-s − 0.770·47-s − 0.672·49-s − 1.53·51-s + 0.0213·53-s + 0.573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.236337349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.236337349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 53 | \( 1 - 0.155T + 53T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 + 4.49T + 61T^{2} \) |
| 67 | \( 1 + 7.43T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 0.310T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 - 0.719T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−9.087822411746359328239794736006, −8.667195725256952381851385066459, −7.960933303078214418742344502371, −7.46480571247806026777822324476, −6.46217395749173829786352316714, −4.99471434604086188368997042762, −4.27712431237261146028795606149, −3.19299828891731030066194010586, −2.61295929268592052051967635369, −1.34524408198655258851606060564,
1.34524408198655258851606060564, 2.61295929268592052051967635369, 3.19299828891731030066194010586, 4.27712431237261146028795606149, 4.99471434604086188368997042762, 6.46217395749173829786352316714, 7.46480571247806026777822324476, 7.960933303078214418742344502371, 8.667195725256952381851385066459, 9.087822411746359328239794736006
