| L(s) = 1 | − 3-s + 2.09·5-s − 1.11·7-s + 9-s − 3.05·11-s − 6.80·13-s − 2.09·15-s + 6.88·17-s − 6.06·19-s + 1.11·21-s − 0.508·23-s − 0.623·25-s − 27-s − 4.52·29-s + 1.62·31-s + 3.05·33-s − 2.34·35-s + 4.52·37-s + 6.80·39-s + 3.40·41-s − 7.55·43-s + 2.09·45-s + 12.9·47-s − 5.74·49-s − 6.88·51-s + 5.82·53-s − 6.39·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.935·5-s − 0.423·7-s + 0.333·9-s − 0.922·11-s − 1.88·13-s − 0.540·15-s + 1.66·17-s − 1.39·19-s + 0.244·21-s − 0.105·23-s − 0.124·25-s − 0.192·27-s − 0.840·29-s + 0.291·31-s + 0.532·33-s − 0.396·35-s + 0.744·37-s + 1.09·39-s + 0.531·41-s − 1.15·43-s + 0.311·45-s + 1.89·47-s − 0.820·49-s − 0.964·51-s + 0.800·53-s − 0.862·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208371921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208371921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 6.80T + 13T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 + 0.508T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 - 7.98T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 7.90T + 67T^{2} \) |
| 71 | \( 1 - 3.42T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 4.06T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.77784083342299600113467106272, −7.48247846682177705788562298545, −6.51050001114915027187257753151, −5.87852217667309090571053042663, −5.29096764189060074805078010287, −4.71284021966117998849612439353, −3.63089395673223638479596461208, −2.54190025187079191516241932950, −2.02762157373263447637665431106, −0.56562823562890893731280143581,
0.56562823562890893731280143581, 2.02762157373263447637665431106, 2.54190025187079191516241932950, 3.63089395673223638479596461208, 4.71284021966117998849612439353, 5.29096764189060074805078010287, 5.87852217667309090571053042663, 6.51050001114915027187257753151, 7.48247846682177705788562298545, 7.77784083342299600113467106272
