| L(s) = 1 | − 3·5-s + 2·7-s + 11-s + 6·17-s + 4·19-s + 23-s + 4·25-s − 8·29-s + 7·31-s − 6·35-s + 37-s − 4·41-s + 6·43-s − 8·47-s − 3·49-s + 2·53-s − 3·55-s + 59-s − 4·61-s − 5·67-s + 3·71-s + 16·73-s + 2·77-s − 2·79-s + 2·83-s − 18·85-s − 15·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s + 0.301·11-s + 1.45·17-s + 0.917·19-s + 0.208·23-s + 4/5·25-s − 1.48·29-s + 1.25·31-s − 1.01·35-s + 0.164·37-s − 0.624·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.404·55-s + 0.130·59-s − 0.512·61-s − 0.610·67-s + 0.356·71-s + 1.87·73-s + 0.227·77-s − 0.225·79-s + 0.219·83-s − 1.95·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736087912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736087912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.927629123853046774697751456251, −7.55252528081176974112132450170, −6.82671507782272554192070208796, −5.79242088578546656641015440150, −5.10583765748157113159455391169, −4.37221035310191883411362450637, −3.60822760858280223727828451956, −3.02374111947685053319903637317, −1.66294893612641948046386286042, −0.72016671080523259380610994522,
0.72016671080523259380610994522, 1.66294893612641948046386286042, 3.02374111947685053319903637317, 3.60822760858280223727828451956, 4.37221035310191883411362450637, 5.10583765748157113159455391169, 5.79242088578546656641015440150, 6.82671507782272554192070208796, 7.55252528081176974112132450170, 7.927629123853046774697751456251
