| L(s) = 1 | − 9·3-s + 24·5-s − 49·7-s + 81·9-s − 66·11-s + 98·13-s − 216·15-s − 216·17-s + 340·19-s + 441·21-s + 1.03e3·23-s − 2.54e3·25-s − 729·27-s − 2.49e3·29-s + 7.04e3·31-s + 594·33-s − 1.17e3·35-s − 1.22e4·37-s − 882·39-s + 6.46e3·41-s + 1.54e4·43-s + 1.94e3·45-s − 2.06e4·47-s + 2.40e3·49-s + 1.94e3·51-s + 3.24e4·53-s − 1.58e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.429·5-s − 0.377·7-s + 1/3·9-s − 0.164·11-s + 0.160·13-s − 0.247·15-s − 0.181·17-s + 0.216·19-s + 0.218·21-s + 0.409·23-s − 0.815·25-s − 0.192·27-s − 0.549·29-s + 1.31·31-s + 0.0949·33-s − 0.162·35-s − 1.46·37-s − 0.0928·39-s + 0.600·41-s + 1.27·43-s + 0.143·45-s − 1.36·47-s + 1/7·49-s + 0.104·51-s + 1.58·53-s − 0.0706·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.570594118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570594118\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 24 T + p^{5} T^{2} \) |
| 11 | \( 1 + 6 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 98 T + p^{5} T^{2} \) |
| 17 | \( 1 + 216 T + p^{5} T^{2} \) |
| 19 | \( 1 - 340 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1038 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2490 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7048 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12238 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6468 T + p^{5} T^{2} \) |
| 43 | \( 1 - 15412 T + p^{5} T^{2} \) |
| 47 | \( 1 + 20604 T + p^{5} T^{2} \) |
| 53 | \( 1 - 32490 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34224 T + p^{5} T^{2} \) |
| 61 | \( 1 - 35654 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12680 T + p^{5} T^{2} \) |
| 71 | \( 1 - 42642 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33734 T + p^{5} T^{2} \) |
| 79 | \( 1 - 85108 T + p^{5} T^{2} \) |
| 83 | \( 1 - 106764 T + p^{5} T^{2} \) |
| 89 | \( 1 - 34884 T + p^{5} T^{2} \) |
| 97 | \( 1 - 18662 T + p^{5} T^{2} \) |
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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
−10.68713333288016604261300323906, −9.888918160455992619229168651973, −9.012313614712693100519160442273, −7.79356508267304610060163563719, −6.71098295210445021640023294193, −5.87127380338154036328048077131, −4.88927677860030034183450172715, −3.58411228101088867263840724030, −2.14830011079263387571484286844, −0.70505553769812684164230064736,
0.70505553769812684164230064736, 2.14830011079263387571484286844, 3.58411228101088867263840724030, 4.88927677860030034183450172715, 5.87127380338154036328048077131, 6.71098295210445021640023294193, 7.79356508267304610060163563719, 9.012313614712693100519160442273, 9.888918160455992619229168651973, 10.68713333288016604261300323906
