| L(s) = 1 | + 2-s + 3-s + 4-s + 0.137·5-s + 6-s + 3.14·7-s + 8-s + 9-s + 0.137·10-s + 1.47·11-s + 12-s − 1.75·13-s + 3.14·14-s + 0.137·15-s + 16-s − 4.07·17-s + 18-s + 1.58·19-s + 0.137·20-s + 3.14·21-s + 1.47·22-s + 8.54·23-s + 24-s − 4.98·25-s − 1.75·26-s + 27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0613·5-s + 0.408·6-s + 1.18·7-s + 0.353·8-s + 0.333·9-s + 0.0433·10-s + 0.445·11-s + 0.288·12-s − 0.487·13-s + 0.839·14-s + 0.0354·15-s + 0.250·16-s − 0.989·17-s + 0.235·18-s + 0.364·19-s + 0.0306·20-s + 0.685·21-s + 0.314·22-s + 1.78·23-s + 0.204·24-s − 0.996·25-s − 0.344·26-s + 0.192·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.662794529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.662794529\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 0.137T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 1.75T + 13T^{2} \) |
| 17 | \( 1 + 4.07T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 - 8.54T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 - 0.711T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 3.50T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 + 0.0537T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 + 1.11T + 71T^{2} \) |
| 73 | \( 1 - 7.44T + 73T^{2} \) |
| 79 | \( 1 + 5.85T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−9.482151478271728627546117509300, −8.797072599136514013813654424228, −7.894329022570576977701153824573, −7.20404994181485955035389662166, −6.33758118977288126870524415542, −5.10118056948758892882713869020, −4.62173370353024827775953940146, −3.56354993593256664956988032635, −2.46649901818912267667727627398, −1.47676215755220921337298441505,
1.47676215755220921337298441505, 2.46649901818912267667727627398, 3.56354993593256664956988032635, 4.62173370353024827775953940146, 5.10118056948758892882713869020, 6.33758118977288126870524415542, 7.20404994181485955035389662166, 7.894329022570576977701153824573, 8.797072599136514013813654424228, 9.482151478271728627546117509300
