| L(s) = 1 | + 3·2-s − 3·3-s + 4-s − 9·6-s + 16·7-s − 21·8-s + 9·9-s − 36·11-s − 3·12-s − 13·13-s + 48·14-s − 71·16-s + 30·17-s + 27·18-s + 68·19-s − 48·21-s − 108·22-s + 120·23-s + 63·24-s − 39·26-s − 27·27-s + 16·28-s − 186·29-s + 8·31-s − 45·32-s + 108·33-s + 90·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 1/8·4-s − 0.612·6-s + 0.863·7-s − 0.928·8-s + 1/3·9-s − 0.986·11-s − 0.0721·12-s − 0.277·13-s + 0.916·14-s − 1.10·16-s + 0.428·17-s + 0.353·18-s + 0.821·19-s − 0.498·21-s − 1.04·22-s + 1.08·23-s + 0.535·24-s − 0.294·26-s − 0.192·27-s + 0.107·28-s − 1.19·29-s + 0.0463·31-s − 0.248·32-s + 0.569·33-s + 0.453·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.528881602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.528881602\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 342 T + p^{3} T^{2} \) |
| 43 | \( 1 - 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 552 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 - 780 T + p^{3} T^{2} \) |
| 61 | \( 1 + 154 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1056 T + p^{3} T^{2} \) |
| 73 | \( 1 - 22 T + p^{3} T^{2} \) |
| 79 | \( 1 + 112 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 90 T + p^{3} T^{2} \) |
| 97 | \( 1 - 334 T + p^{3} T^{2} \) |
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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
−9.727710906436437382911283066487, −8.784834428323512021797984673283, −7.75832040454384126612603830927, −6.98682752324370834169318554572, −5.66683843991386950987619936357, −5.30713250403921430146680790561, −4.54868894058659111206792154583, −3.48628185209345796580726529955, −2.34104331960217180849058132348, −0.74828824501741003395328360003,
0.74828824501741003395328360003, 2.34104331960217180849058132348, 3.48628185209345796580726529955, 4.54868894058659111206792154583, 5.30713250403921430146680790561, 5.66683843991386950987619936357, 6.98682752324370834169318554572, 7.75832040454384126612603830927, 8.784834428323512021797984673283, 9.727710906436437382911283066487
