| L(s) = 1 | − 2.58·2-s − 3-s + 4.67·4-s − 0.798·5-s + 2.58·6-s − 6.91·8-s + 9-s + 2.06·10-s − 1.79·11-s − 4.67·12-s + 3.77·13-s + 0.798·15-s + 8.52·16-s − 3.97·17-s − 2.58·18-s + 0.225·19-s − 3.73·20-s + 4.63·22-s + 5.53·23-s + 6.91·24-s − 4.36·25-s − 9.74·26-s − 27-s + 2.98·29-s − 2.06·30-s − 3.70·31-s − 8.18·32-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.33·4-s − 0.357·5-s + 1.05·6-s − 2.44·8-s + 0.333·9-s + 0.652·10-s − 0.540·11-s − 1.35·12-s + 1.04·13-s + 0.206·15-s + 2.13·16-s − 0.965·17-s − 0.609·18-s + 0.0516·19-s − 0.835·20-s + 0.988·22-s + 1.15·23-s + 1.41·24-s − 0.872·25-s − 1.91·26-s − 0.192·27-s + 0.553·29-s − 0.376·30-s − 0.664·31-s − 1.44·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4521542456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4521542456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 + 0.798T + 5T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 3.97T + 17T^{2} \) |
| 19 | \( 1 - 0.225T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 5.75T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 - 7.82T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 0.117T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 - 2.28T + 89T^{2} \) |
| 97 | \( 1 - 0.883T + 97T^{2} \) |
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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
−8.031597059173522538540482331919, −7.67368368505027701511291760724, −6.72575403560689847085383730281, −6.38249324370773795737623965746, −5.47555147000377075776520725042, −4.47055144521285529302591107824, −3.38667089326560310720633728720, −2.40414395062902847277580028980, −1.44817277885100953782920977850, −0.50497768980381949016395208320,
0.50497768980381949016395208320, 1.44817277885100953782920977850, 2.40414395062902847277580028980, 3.38667089326560310720633728720, 4.47055144521285529302591107824, 5.47555147000377075776520725042, 6.38249324370773795737623965746, 6.72575403560689847085383730281, 7.67368368505027701511291760724, 8.031597059173522538540482331919
