| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 6·17-s + 4·19-s − 6·21-s + 23-s − 9·27-s + 8·29-s + 7·31-s + 37-s − 4·41-s − 6·43-s − 8·47-s − 3·49-s + 18·51-s − 2·53-s − 12·57-s + 59-s − 4·61-s + 12·63-s − 5·67-s − 3·69-s − 3·71-s + 16·73-s + 2·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 1.45·17-s + 0.917·19-s − 1.30·21-s + 0.208·23-s − 1.73·27-s + 1.48·29-s + 1.25·31-s + 0.164·37-s − 0.624·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s + 2.52·51-s − 0.274·53-s − 1.58·57-s + 0.130·59-s − 0.512·61-s + 1.51·63-s − 0.610·67-s − 0.361·69-s − 0.356·71-s + 1.87·73-s + 0.225·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146821930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146821930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.66531287273229, −13.81023096913344, −13.58234886731371, −12.90637278013464, −12.31444775071519, −11.86743425434779, −11.36985481292664, −11.22474928394209, −10.43720337018976, −10.15278160780374, −9.508687106955276, −8.759161408677722, −8.183911790583010, −7.602916338900123, −6.858380707972840, −6.437556124268058, −6.129002066510711, −5.147290361179847, −4.844234193319321, −4.601458905630860, −3.672816312916084, −2.794998215873005, −1.872554294327303, −1.194772263599797, −0.4740064361685214,
0.4740064361685214, 1.194772263599797, 1.872554294327303, 2.794998215873005, 3.672816312916084, 4.601458905630860, 4.844234193319321, 5.147290361179847, 6.129002066510711, 6.437556124268058, 6.858380707972840, 7.602916338900123, 8.183911790583010, 8.759161408677722, 9.508687106955276, 10.15278160780374, 10.43720337018976, 11.22474928394209, 11.36985481292664, 11.86743425434779, 12.31444775071519, 12.90637278013464, 13.58234886731371, 13.81023096913344, 14.66531287273229
