| L(s) = 1 | + 4·5-s + 4·7-s − 2·11-s − 13-s + 6·17-s − 4·19-s + 4·23-s + 11·25-s + 6·29-s − 8·31-s + 16·35-s − 10·37-s + 4·41-s + 4·43-s − 6·47-s + 9·49-s − 6·53-s − 8·55-s − 6·59-s − 6·61-s − 4·65-s + 10·71-s − 2·73-s − 8·77-s − 10·83-s + 24·85-s − 8·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s − 1.43·31-s + 2.70·35-s − 1.64·37-s + 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.07·55-s − 0.781·59-s − 0.768·61-s − 0.496·65-s + 1.18·71-s − 0.234·73-s − 0.911·77-s − 1.09·83-s + 2.60·85-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.893318383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.893318383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.236517015935069263593512559649, −8.511713783286943073331410960949, −7.72620902983739830530194611782, −6.83090920971029639184526540146, −5.79542508643660039855358485463, −5.26738993533513128275686270358, −4.64540211102710374344620461150, −3.07082311219214790751277479667, −2.04771532659732672600710243609, −1.35061874417219692087547083371,
1.35061874417219692087547083371, 2.04771532659732672600710243609, 3.07082311219214790751277479667, 4.64540211102710374344620461150, 5.26738993533513128275686270358, 5.79542508643660039855358485463, 6.83090920971029639184526540146, 7.72620902983739830530194611782, 8.511713783286943073331410960949, 9.236517015935069263593512559649
