| L(s) = 1 | − 0.593·5-s − 2.05·7-s − 3.05·11-s − 4.05·13-s − 17-s − 19-s + 4.32·23-s − 4.64·25-s + 0.672·29-s + 9.97·31-s + 1.21·35-s + 0.460·37-s − 9.24·41-s + 4.32·43-s − 10.8·47-s − 2.78·49-s + 1.46·53-s + 1.81·55-s − 2.19·59-s − 3.13·61-s + 2.40·65-s − 10.1·67-s − 5.16·71-s − 11.9·73-s + 6.27·77-s − 5.75·79-s + 11.6·83-s + ⋯ |
| L(s) = 1 | − 0.265·5-s − 0.776·7-s − 0.920·11-s − 1.12·13-s − 0.242·17-s − 0.229·19-s + 0.902·23-s − 0.929·25-s + 0.124·29-s + 1.79·31-s + 0.206·35-s + 0.0757·37-s − 1.44·41-s + 0.659·43-s − 1.58·47-s − 0.397·49-s + 0.200·53-s + 0.244·55-s − 0.285·59-s − 0.401·61-s + 0.298·65-s − 1.23·67-s − 0.612·71-s − 1.39·73-s + 0.714·77-s − 0.647·79-s + 1.27·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8199050476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8199050476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 0.593T + 5T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 - 0.672T + 29T^{2} \) |
| 31 | \( 1 - 9.97T + 31T^{2} \) |
| 37 | \( 1 - 0.460T + 37T^{2} \) |
| 41 | \( 1 + 9.24T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.78339443903544667889433065473, −7.14740958617273137262728095988, −6.48491742965871734400358888247, −5.79244674344484149086601559416, −4.85745774214389739645662467389, −4.48276550307790261269124223796, −3.25575841104521945125694560252, −2.84513219441006407483681277860, −1.86765569462993955403144463481, −0.42249642226584963529028688612,
0.42249642226584963529028688612, 1.86765569462993955403144463481, 2.84513219441006407483681277860, 3.25575841104521945125694560252, 4.48276550307790261269124223796, 4.85745774214389739645662467389, 5.79244674344484149086601559416, 6.48491742965871734400358888247, 7.14740958617273137262728095988, 7.78339443903544667889433065473
