| L(s) = 1 | + 3·3-s − 12·5-s + 9·9-s + 60·11-s − 44·13-s − 36·15-s + 128·17-s + 52·19-s + 160·23-s + 19·25-s + 27·27-s − 230·29-s + 136·31-s + 180·33-s − 318·37-s − 132·39-s + 192·41-s − 220·43-s − 108·45-s + 184·47-s + 384·51-s − 498·53-s − 720·55-s + 156·57-s − 492·59-s − 20·61-s + 528·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.07·5-s + 1/3·9-s + 1.64·11-s − 0.938·13-s − 0.619·15-s + 1.82·17-s + 0.627·19-s + 1.45·23-s + 0.151·25-s + 0.192·27-s − 1.47·29-s + 0.787·31-s + 0.949·33-s − 1.41·37-s − 0.541·39-s + 0.731·41-s − 0.780·43-s − 0.357·45-s + 0.571·47-s + 1.05·51-s − 1.29·53-s − 1.76·55-s + 0.362·57-s − 1.08·59-s − 0.0419·61-s + 1.00·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.618613257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.618613257\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 128 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 160 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 318 T + p^{3} T^{2} \) |
| 41 | \( 1 - 192 T + p^{3} T^{2} \) |
| 43 | \( 1 + 220 T + p^{3} T^{2} \) |
| 47 | \( 1 - 184 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 20 T + p^{3} T^{2} \) |
| 67 | \( 1 + 380 T + p^{3} T^{2} \) |
| 71 | \( 1 - 264 T + p^{3} T^{2} \) |
| 73 | \( 1 - 560 T + p^{3} T^{2} \) |
| 79 | \( 1 + 104 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1144 T + p^{3} T^{2} \) |
| 97 | \( 1 - 904 T + p^{3} T^{2} \) |
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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
−8.659614095250369876537065324496, −7.61237860822053760469594499572, −7.45307362883762776350961118697, −6.50283372207163588773195873838, −5.35877495002899977279235462018, −4.50487866002801351002881382120, −3.53314308366900664691506129300, −3.19982074976808305745334317210, −1.68589695001136903612348370952, −0.73568608526238263893635154343,
0.73568608526238263893635154343, 1.68589695001136903612348370952, 3.19982074976808305745334317210, 3.53314308366900664691506129300, 4.50487866002801351002881382120, 5.35877495002899977279235462018, 6.50283372207163588773195873838, 7.45307362883762776350961118697, 7.61237860822053760469594499572, 8.659614095250369876537065324496
