| L(s) = 1 | − 2·3-s + 3·7-s + 9-s + 5·11-s + 4·13-s + 3·17-s − 19-s − 6·21-s − 8·23-s + 4·27-s − 2·29-s + 4·31-s − 10·33-s − 10·37-s − 8·39-s + 10·41-s − 43-s + 47-s + 2·49-s − 6·51-s + 4·53-s + 2·57-s + 6·59-s − 13·61-s + 3·63-s + 12·67-s + 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.727·17-s − 0.229·19-s − 1.30·21-s − 1.66·23-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 1.74·33-s − 1.64·37-s − 1.28·39-s + 1.56·41-s − 0.152·43-s + 0.145·47-s + 2/7·49-s − 0.840·51-s + 0.549·53-s + 0.264·57-s + 0.781·59-s − 1.66·61-s + 0.377·63-s + 1.46·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.489784422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.489784422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.142669581834378698595686186454, −8.426347554243774025274773727164, −7.67036876906477496457872273315, −6.54496232916405090183568591347, −6.05356181637470555300777284468, −5.29796420868215321731869729709, −4.34576334774572475342173538720, −3.62036505864368699268002729080, −1.86326488698354633791416853460, −0.942787934452988901616816885952,
0.942787934452988901616816885952, 1.86326488698354633791416853460, 3.62036505864368699268002729080, 4.34576334774572475342173538720, 5.29796420868215321731869729709, 6.05356181637470555300777284468, 6.54496232916405090183568591347, 7.67036876906477496457872273315, 8.426347554243774025274773727164, 9.142669581834378698595686186454
