| L(s) = 1 | − 2-s + 0.245·3-s + 4-s + 3.30·5-s − 0.245·6-s − 1.69·7-s − 8-s − 2.93·9-s − 3.30·10-s − 4.16·11-s + 0.245·12-s + 5.79·13-s + 1.69·14-s + 0.811·15-s + 16-s − 6.44·17-s + 2.93·18-s − 0.665·19-s + 3.30·20-s − 0.415·21-s + 4.16·22-s + 7.32·23-s − 0.245·24-s + 5.90·25-s − 5.79·26-s − 1.46·27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.141·3-s + 0.5·4-s + 1.47·5-s − 0.100·6-s − 0.638·7-s − 0.353·8-s − 0.979·9-s − 1.04·10-s − 1.25·11-s + 0.0709·12-s + 1.60·13-s + 0.451·14-s + 0.209·15-s + 0.250·16-s − 1.56·17-s + 0.692·18-s − 0.152·19-s + 0.738·20-s − 0.0907·21-s + 0.887·22-s + 1.52·23-s − 0.0502·24-s + 1.18·25-s − 1.13·26-s − 0.281·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473726808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473726808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 0.245T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 0.665T + 19T^{2} \) |
| 23 | \( 1 - 7.32T + 23T^{2} \) |
| 29 | \( 1 + 6.76T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 + 4.29T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 5.76T + 67T^{2} \) |
| 71 | \( 1 - 7.17T + 71T^{2} \) |
| 73 | \( 1 - 2.23T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 - 0.251T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.774320892947258220763773823540, −7.88024890032823123665397093470, −6.87708999971201868686698514395, −6.15824973358413898464528674215, −5.80603310640785921229600252943, −4.92244844206364586032406977099, −3.52417392302436643970969971360, −2.61535022463566515667038399084, −2.11382085113121343099437036083, −0.74343307231476860346074611653,
0.74343307231476860346074611653, 2.11382085113121343099437036083, 2.61535022463566515667038399084, 3.52417392302436643970969971360, 4.92244844206364586032406977099, 5.80603310640785921229600252943, 6.15824973358413898464528674215, 6.87708999971201868686698514395, 7.88024890032823123665397093470, 8.774320892947258220763773823540
