| L(s) = 1 | + 4·2-s + 6·3-s + 16·4-s + 24·6-s + 27·7-s + 64·8-s − 207·9-s − 323·11-s + 96·12-s + 676·13-s + 108·14-s + 256·16-s + 1.10e3·17-s − 828·18-s − 361·19-s + 162·21-s − 1.29e3·22-s − 1.38e3·23-s + 384·24-s + 2.70e3·26-s − 2.70e3·27-s + 432·28-s + 2.87e3·29-s + 1.37e3·31-s + 1.02e3·32-s − 1.93e3·33-s + 4.42e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s + 0.272·6-s + 0.208·7-s + 0.353·8-s − 0.851·9-s − 0.804·11-s + 0.192·12-s + 1.10·13-s + 0.147·14-s + 1/4·16-s + 0.929·17-s − 0.602·18-s − 0.229·19-s + 0.0801·21-s − 0.569·22-s − 0.545·23-s + 0.136·24-s + 0.784·26-s − 0.712·27-s + 0.104·28-s + 0.633·29-s + 0.256·31-s + 0.176·32-s − 0.309·33-s + 0.656·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.180353517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.180353517\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 27 T + p^{5} T^{2} \) |
| 11 | \( 1 + 323 T + p^{5} T^{2} \) |
| 13 | \( 1 - 4 p^{2} T + p^{5} T^{2} \) |
| 17 | \( 1 - 1107 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1384 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2870 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1372 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7982 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1202 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9911 T + p^{5} T^{2} \) |
| 47 | \( 1 + 3463 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17764 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27270 T + p^{5} T^{2} \) |
| 61 | \( 1 - 20867 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15228 T + p^{5} T^{2} \) |
| 71 | \( 1 - 40642 T + p^{5} T^{2} \) |
| 73 | \( 1 - 66711 T + p^{5} T^{2} \) |
| 79 | \( 1 - 68960 T + p^{5} T^{2} \) |
| 83 | \( 1 - 12396 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41220 T + p^{5} T^{2} \) |
| 97 | \( 1 - 113432 T + p^{5} T^{2} \) |
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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
−9.260930864405979160992839764597, −8.138924332085274549464597983417, −7.912135849669399669101218315600, −6.51190499478729370618204767201, −5.79795035942049960499817145185, −4.98898429615969314029908079002, −3.84034182580371156053834777727, −3.03878091788683589431245925458, −2.12131664261860847138507996046, −0.78883911633374194384538526888,
0.78883911633374194384538526888, 2.12131664261860847138507996046, 3.03878091788683589431245925458, 3.84034182580371156053834777727, 4.98898429615969314029908079002, 5.79795035942049960499817145185, 6.51190499478729370618204767201, 7.912135849669399669101218315600, 8.138924332085274549464597983417, 9.260930864405979160992839764597
