| L(s) = 1 | − 5·3-s − 5·5-s + 19·7-s − 2·9-s + 11·11-s − 62·13-s + 25·15-s + 19·17-s + 131·19-s − 95·21-s − 138·23-s + 25·25-s + 145·27-s − 79·29-s − 217·31-s − 55·33-s − 95·35-s − 91·37-s + 310·39-s + 158·41-s − 120·43-s + 10·45-s + 546·47-s + 18·49-s − 95·51-s − 439·53-s − 55·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.447·5-s + 1.02·7-s − 0.0740·9-s + 0.301·11-s − 1.32·13-s + 0.430·15-s + 0.271·17-s + 1.58·19-s − 0.987·21-s − 1.25·23-s + 1/5·25-s + 1.03·27-s − 0.505·29-s − 1.25·31-s − 0.290·33-s − 0.458·35-s − 0.404·37-s + 1.27·39-s + 0.601·41-s − 0.425·43-s + 0.0331·45-s + 1.69·47-s + 0.0524·49-s − 0.260·51-s − 1.13·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.084449903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084449903\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 19 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 19 T + p^{3} T^{2} \) |
| 19 | \( 1 - 131 T + p^{3} T^{2} \) |
| 23 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 79 T + p^{3} T^{2} \) |
| 31 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 91 T + p^{3} T^{2} \) |
| 41 | \( 1 - 158 T + p^{3} T^{2} \) |
| 43 | \( 1 + 120 T + p^{3} T^{2} \) |
| 47 | \( 1 - 546 T + p^{3} T^{2} \) |
| 53 | \( 1 + 439 T + p^{3} T^{2} \) |
| 59 | \( 1 + 290 T + p^{3} T^{2} \) |
| 61 | \( 1 + 373 T + p^{3} T^{2} \) |
| 67 | \( 1 + 728 T + p^{3} T^{2} \) |
| 71 | \( 1 - 709 T + p^{3} T^{2} \) |
| 73 | \( 1 - 850 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1194 T + p^{3} T^{2} \) |
| 83 | \( 1 + 58 T + p^{3} T^{2} \) |
| 89 | \( 1 - 753 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1228 T + p^{3} T^{2} \) |
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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
−9.836622013996962648478522547001, −8.969182770847173677289339502915, −7.73321114548200445877947686347, −7.43204309406659689757914207905, −6.12265291175672629376249954486, −5.26745254378133293429576691517, −4.68512921578643237566216426426, −3.42523291305374735306006562610, −1.95981101313403247659848500401, −0.59504354255887790366406273772,
0.59504354255887790366406273772, 1.95981101313403247659848500401, 3.42523291305374735306006562610, 4.68512921578643237566216426426, 5.26745254378133293429576691517, 6.12265291175672629376249954486, 7.43204309406659689757914207905, 7.73321114548200445877947686347, 8.969182770847173677289339502915, 9.836622013996962648478522547001
