| L(s) = 1 | + 3-s + 9-s − 2·11-s − 5·13-s + 2·17-s + 3·19-s − 2·23-s − 5·25-s + 27-s + 8·29-s − 31-s − 2·33-s − 5·37-s − 5·39-s − 2·41-s − 7·43-s − 8·47-s + 2·51-s − 2·53-s + 3·57-s + 10·59-s + 2·61-s + 11·67-s − 2·69-s − 12·71-s + 3·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 0.485·17-s + 0.688·19-s − 0.417·23-s − 25-s + 0.192·27-s + 1.48·29-s − 0.179·31-s − 0.348·33-s − 0.821·37-s − 0.800·39-s − 0.312·41-s − 1.06·43-s − 1.16·47-s + 0.280·51-s − 0.274·53-s + 0.397·57-s + 1.30·59-s + 0.256·61-s + 1.34·67-s − 0.240·69-s − 1.42·71-s + 0.351·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.048191487728243322145007863091, −7.26659301468958733121166374062, −6.71270102690899525841442572868, −5.56540415156406552357106998644, −5.02645578918039541898832978816, −4.15299607182744035112334853502, −3.17772620331725117881503369395, −2.53396280729970689748405554083, −1.53446718549130139861009880797, 0,
1.53446718549130139861009880797, 2.53396280729970689748405554083, 3.17772620331725117881503369395, 4.15299607182744035112334853502, 5.02645578918039541898832978816, 5.56540415156406552357106998644, 6.71270102690899525841442572868, 7.26659301468958733121166374062, 8.048191487728243322145007863091
