| L(s) = 1 | + 3-s − 2.81·7-s + 9-s − 11-s − 7.06·13-s − 0.816·17-s − 7.88·19-s − 2.81·21-s + 5·23-s + 27-s + 7.06·29-s − 6.70·31-s − 33-s + 7.70·37-s − 7.06·39-s − 6.81·41-s + 8.70·43-s + 8.06·47-s + 0.931·49-s − 0.816·51-s + 9.63·53-s − 7.88·57-s + 2.06·59-s − 2·61-s − 2.81·63-s − 11.7·67-s + 5·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.06·7-s + 0.333·9-s − 0.301·11-s − 1.96·13-s − 0.197·17-s − 1.80·19-s − 0.614·21-s + 1.04·23-s + 0.192·27-s + 1.31·29-s − 1.20·31-s − 0.174·33-s + 1.26·37-s − 1.13·39-s − 1.06·41-s + 1.32·43-s + 1.17·47-s + 0.133·49-s − 0.114·51-s + 1.32·53-s − 1.04·57-s + 0.269·59-s − 0.256·61-s − 0.354·63-s − 1.43·67-s + 0.601·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.386833250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386833250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2.81T + 7T^{2} \) |
| 13 | \( 1 + 7.06T + 13T^{2} \) |
| 17 | \( 1 + 0.816T + 17T^{2} \) |
| 19 | \( 1 + 7.88T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 0.632T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.95T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.892363140096563500681607908874, −7.30281666954975948288482235334, −6.70971703215664497554871450389, −6.02205574100866275005547936957, −4.97893373149295734559776625609, −4.41943587048010234737862627925, −3.49727302040902878768311962249, −2.58121915047586747897086918079, −2.24179911709613846635696661671, −0.55103969662514905842763736489,
0.55103969662514905842763736489, 2.24179911709613846635696661671, 2.58121915047586747897086918079, 3.49727302040902878768311962249, 4.41943587048010234737862627925, 4.97893373149295734559776625609, 6.02205574100866275005547936957, 6.70971703215664497554871450389, 7.30281666954975948288482235334, 7.892363140096563500681607908874
