| L(s) = 1 | − 3·3-s + 15·5-s + 9·9-s + 9·11-s + 88·13-s − 45·15-s + 84·17-s + 104·19-s + 84·23-s + 100·25-s − 27·27-s + 51·29-s + 185·31-s − 27·33-s + 44·37-s − 264·39-s + 168·41-s − 326·43-s + 135·45-s − 138·47-s − 252·51-s + 639·53-s + 135·55-s − 312·57-s + 159·59-s − 722·61-s + 1.32e3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.246·11-s + 1.87·13-s − 0.774·15-s + 1.19·17-s + 1.25·19-s + 0.761·23-s + 4/5·25-s − 0.192·27-s + 0.326·29-s + 1.07·31-s − 0.142·33-s + 0.195·37-s − 1.08·39-s + 0.639·41-s − 1.15·43-s + 0.447·45-s − 0.428·47-s − 0.691·51-s + 1.65·53-s + 0.330·55-s − 0.725·57-s + 0.350·59-s − 1.51·61-s + 2.51·65-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
| \(L(2)\) |
\(\approx\) |
\(3.495252911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.495252911\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 88 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 185 T + p^{3} T^{2} \) |
| 37 | \( 1 - 44 T + p^{3} T^{2} \) |
| 41 | \( 1 - 168 T + p^{3} T^{2} \) |
| 43 | \( 1 + 326 T + p^{3} T^{2} \) |
| 47 | \( 1 + 138 T + p^{3} T^{2} \) |
| 53 | \( 1 - 639 T + p^{3} T^{2} \) |
| 59 | \( 1 - 159 T + p^{3} T^{2} \) |
| 61 | \( 1 + 722 T + p^{3} T^{2} \) |
| 67 | \( 1 - 166 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 583 T + p^{3} T^{2} \) |
| 83 | \( 1 + 597 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 169 T + p^{3} T^{2} \) |
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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
−8.774896001561766973573570655313, −7.86884999126737567132525331475, −6.86105820313880743775225600064, −6.10695225573573362196239254945, −5.68351060504712338838198260569, −4.90616185071500007109458417129, −3.70258767122522636805247927013, −2.83173955222539299078340733956, −1.40893490864760122187428798038, −1.03556745157569230889580165492,
1.03556745157569230889580165492, 1.40893490864760122187428798038, 2.83173955222539299078340733956, 3.70258767122522636805247927013, 4.90616185071500007109458417129, 5.68351060504712338838198260569, 6.10695225573573362196239254945, 6.86105820313880743775225600064, 7.86884999126737567132525331475, 8.774896001561766973573570655313
