| L(s) = 1 | − 7·3-s − 7·7-s + 22·9-s − 39·11-s + 13·13-s + 24·17-s − 38·19-s + 49·21-s − 39·23-s − 125·25-s + 35·27-s − 96·29-s − 227·31-s + 273·33-s + 425·37-s − 91·39-s − 105·41-s − 344·43-s − 99·47-s + 49·49-s − 168·51-s − 540·53-s + 266·57-s − 114·59-s − 565·61-s − 154·63-s + 385·67-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.377·7-s + 0.814·9-s − 1.06·11-s + 0.277·13-s + 0.342·17-s − 0.458·19-s + 0.509·21-s − 0.353·23-s − 25-s + 0.249·27-s − 0.614·29-s − 1.31·31-s + 1.44·33-s + 1.88·37-s − 0.373·39-s − 0.399·41-s − 1.21·43-s − 0.307·47-s + 1/7·49-s − 0.461·51-s − 1.39·53-s + 0.618·57-s − 0.251·59-s − 1.18·61-s − 0.307·63-s + 0.702·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4078178899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4078178899\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 39 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 227 T + p^{3} T^{2} \) |
| 37 | \( 1 - 425 T + p^{3} T^{2} \) |
| 41 | \( 1 + 105 T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 99 T + p^{3} T^{2} \) |
| 53 | \( 1 + 540 T + p^{3} T^{2} \) |
| 59 | \( 1 + 114 T + p^{3} T^{2} \) |
| 61 | \( 1 + 565 T + p^{3} T^{2} \) |
| 67 | \( 1 - 385 T + p^{3} T^{2} \) |
| 71 | \( 1 - 156 T + p^{3} T^{2} \) |
| 73 | \( 1 + 673 T + p^{3} T^{2} \) |
| 79 | \( 1 + 749 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1044 T + p^{3} T^{2} \) |
| 89 | \( 1 + 690 T + p^{3} T^{2} \) |
| 97 | \( 1 - 317 T + p^{3} T^{2} \) |
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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
−9.367332216927822846817480335584, −8.181238676529564166376880012574, −7.49209743245726972345685547106, −6.45263902492938738283475185716, −5.84675832212938222664461021736, −5.19669759247800625343372726739, −4.23735929148356095622898389747, −3.08769932509414204700880004834, −1.76093438267581520132164888550, −0.32818667882405120331016993944,
0.32818667882405120331016993944, 1.76093438267581520132164888550, 3.08769932509414204700880004834, 4.23735929148356095622898389747, 5.19669759247800625343372726739, 5.84675832212938222664461021736, 6.45263902492938738283475185716, 7.49209743245726972345685547106, 8.181238676529564166376880012574, 9.367332216927822846817480335584
