| L(s) = 1 | + 4·3-s − 16·7-s − 11·9-s − 36·11-s − 42·13-s + 110·17-s + 116·19-s − 64·21-s − 16·23-s − 152·27-s − 198·29-s + 240·31-s − 144·33-s − 258·37-s − 168·39-s + 442·41-s − 292·43-s − 392·47-s − 87·49-s + 440·51-s + 142·53-s + 464·57-s + 348·59-s + 570·61-s + 176·63-s + 692·67-s − 64·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.863·7-s − 0.407·9-s − 0.986·11-s − 0.896·13-s + 1.56·17-s + 1.40·19-s − 0.665·21-s − 0.145·23-s − 1.08·27-s − 1.26·29-s + 1.39·31-s − 0.759·33-s − 1.14·37-s − 0.689·39-s + 1.68·41-s − 1.03·43-s − 1.21·47-s − 0.253·49-s + 1.20·51-s + 0.368·53-s + 1.07·57-s + 0.767·59-s + 1.19·61-s + 0.351·63-s + 1.26·67-s − 0.111·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.909479964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909479964\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 110 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 240 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 442 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 392 T + p^{3} T^{2} \) |
| 53 | \( 1 - 142 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 - 692 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 134 T + p^{3} T^{2} \) |
| 79 | \( 1 - 784 T + p^{3} T^{2} \) |
| 83 | \( 1 - 564 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−9.184478487980776786913169372570, −7.996357621438037820976494231465, −7.76209990395498513275104207886, −6.75863107358783973343171306583, −5.60543067389888406670225779166, −5.11336340263271274269846223658, −3.55994626956546019092066885841, −3.09062033749705430836611736344, −2.19512779974675222682689963517, −0.61341272710923081709528458046,
0.61341272710923081709528458046, 2.19512779974675222682689963517, 3.09062033749705430836611736344, 3.55994626956546019092066885841, 5.11336340263271274269846223658, 5.60543067389888406670225779166, 6.75863107358783973343171306583, 7.76209990395498513275104207886, 7.996357621438037820976494231465, 9.184478487980776786913169372570
