| L(s) = 1 | − 2-s + 4-s + 3.23·5-s − 8-s − 3.23·10-s + 11-s + 16-s + 0.763·17-s + 5.70·19-s + 3.23·20-s − 22-s − 6.47·23-s + 5.47·25-s + 4.47·29-s − 7.23·31-s − 32-s − 0.763·34-s + 6.94·37-s − 5.70·38-s − 3.23·40-s + 0.763·41-s − 2.47·43-s + 44-s + 6.47·46-s + 9.70·47-s − 5.47·50-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.353·8-s − 1.02·10-s + 0.301·11-s + 0.250·16-s + 0.185·17-s + 1.30·19-s + 0.723·20-s − 0.213·22-s − 1.34·23-s + 1.09·25-s + 0.830·29-s − 1.29·31-s − 0.176·32-s − 0.131·34-s + 1.14·37-s − 0.925·38-s − 0.511·40-s + 0.119·41-s − 0.376·43-s + 0.150·44-s + 0.954·46-s + 1.41·47-s − 0.773·50-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208161470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208161470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 + 2.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.51094990459684057901788150817, −7.21406646171574540605324781367, −6.07815469823933489417662159348, −5.94958910523081207352979633510, −5.17214062048125914071591313762, −4.16682813317038650899691498016, −3.18832118192243403912499352078, −2.36041988340953305810774911917, −1.68590016565790179826447577587, −0.816454758009094986539579143561,
0.816454758009094986539579143561, 1.68590016565790179826447577587, 2.36041988340953305810774911917, 3.18832118192243403912499352078, 4.16682813317038650899691498016, 5.17214062048125914071591313762, 5.94958910523081207352979633510, 6.07815469823933489417662159348, 7.21406646171574540605324781367, 7.51094990459684057901788150817
