| L(s) = 1 | + 2-s + 3-s + 4-s + 3.10·5-s + 6-s + 7-s + 8-s + 9-s + 3.10·10-s + 1.55·11-s + 12-s + 14-s + 3.10·15-s + 16-s − 1.24·17-s + 18-s − 4.73·19-s + 3.10·20-s + 21-s + 1.55·22-s + 5.10·23-s + 24-s + 4.66·25-s + 27-s + 28-s − 1.05·29-s + 3.10·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.39·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.983·10-s + 0.468·11-s + 0.288·12-s + 0.267·14-s + 0.802·15-s + 0.250·16-s − 0.302·17-s + 0.235·18-s − 1.08·19-s + 0.695·20-s + 0.218·21-s + 0.331·22-s + 1.06·23-s + 0.204·24-s + 0.932·25-s + 0.192·27-s + 0.188·28-s − 0.196·29-s + 0.567·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.053546632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.053546632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.10T + 5T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 7.92T + 37T^{2} \) |
| 41 | \( 1 - 2.71T + 41T^{2} \) |
| 43 | \( 1 + 2.63T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 4.40T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.892495600161855759638537968807, −7.06268255430073328110318853215, −6.41029899867197567243277263481, −5.87688959022457688382903699798, −5.05132280527931610874781366254, −4.41651638248857286797078542450, −3.57335196784353281113865458532, −2.57521706810151933875405724276, −2.07001402417403767085478932875, −1.19392505837996915605495070702,
1.19392505837996915605495070702, 2.07001402417403767085478932875, 2.57521706810151933875405724276, 3.57335196784353281113865458532, 4.41651638248857286797078542450, 5.05132280527931610874781366254, 5.87688959022457688382903699798, 6.41029899867197567243277263481, 7.06268255430073328110318853215, 7.892495600161855759638537968807
