| L(s) = 1 | − 2.49·3-s − 2.69·5-s − 7-s + 3.24·9-s + 3.96·11-s − 0.387·13-s + 6.74·15-s − 17-s − 4.75·19-s + 2.49·21-s − 0.353·23-s + 2.28·25-s − 0.618·27-s − 9.19·29-s + 3.98·31-s − 9.91·33-s + 2.69·35-s + 5.51·37-s + 0.967·39-s − 8.86·41-s − 10.8·43-s − 8.76·45-s + 11.1·47-s + 49-s + 2.49·51-s − 5.13·53-s − 10.7·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s − 1.20·5-s − 0.377·7-s + 1.08·9-s + 1.19·11-s − 0.107·13-s + 1.74·15-s − 0.242·17-s − 1.09·19-s + 0.545·21-s − 0.0737·23-s + 0.457·25-s − 0.118·27-s − 1.70·29-s + 0.716·31-s − 1.72·33-s + 0.456·35-s + 0.906·37-s + 0.154·39-s − 1.38·41-s − 1.64·43-s − 1.30·45-s + 1.62·47-s + 0.142·49-s + 0.349·51-s − 0.705·53-s − 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3907533864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3907533864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 + 0.387T + 13T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 + 0.353T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 8.73T + 73T^{2} \) |
| 79 | \( 1 - 7.03T + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−8.454237865587634691088701061644, −7.61102742352676966019190274352, −6.81581481757065827725694956710, −6.35938098689077236868360343295, −5.61397616102147282796919758556, −4.57350507496847517440348404183, −4.12906106194391031114378340207, −3.24711950465633811094158661376, −1.68402542734146027172521200670, −0.39073425807401728711869271077,
0.39073425807401728711869271077, 1.68402542734146027172521200670, 3.24711950465633811094158661376, 4.12906106194391031114378340207, 4.57350507496847517440348404183, 5.61397616102147282796919758556, 6.35938098689077236868360343295, 6.81581481757065827725694956710, 7.61102742352676966019190274352, 8.454237865587634691088701061644
