| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s − 11-s − 12-s − 5·13-s + 16-s − 6·17-s − 2·18-s − 2·19-s − 22-s + 6·23-s − 24-s − 5·25-s − 5·26-s + 5·27-s + 3·29-s − 8·31-s + 32-s + 33-s − 6·34-s − 2·36-s + 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.301·11-s − 0.288·12-s − 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.471·18-s − 0.458·19-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s − 0.980·26-s + 0.962·27-s + 0.557·29-s − 1.43·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s − 1/3·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−9.534472114696256468976714704500, −8.651602461969610594249352273118, −7.57292839693101241292866908605, −6.78055619630445991016188048984, −5.95863722316307196134578225481, −5.06190011008125552875700991007, −4.45886918939457859110202630280, −3.06901423548217734744257741815, −2.12523769647597520228021232369, 0,
2.12523769647597520228021232369, 3.06901423548217734744257741815, 4.45886918939457859110202630280, 5.06190011008125552875700991007, 5.95863722316307196134578225481, 6.78055619630445991016188048984, 7.57292839693101241292866908605, 8.651602461969610594249352273118, 9.534472114696256468976714704500
