| L(s) = 1 | − 3·3-s + 21.3·5-s + 9·9-s − 31.3·11-s − 84.7·13-s − 64.0·15-s + 16.0·17-s + 20·19-s + 80.7·23-s + 331.·25-s − 27·27-s + 102·29-s − 245.·31-s + 94.0·33-s + 215.·37-s + 254.·39-s − 150.·41-s − 441.·43-s + 192.·45-s − 206.·47-s − 48.2·51-s + 426.·53-s − 669.·55-s − 60·57-s + 363.·59-s + 343.·61-s − 1.80e3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.91·5-s + 0.333·9-s − 0.859·11-s − 1.80·13-s − 1.10·15-s + 0.229·17-s + 0.241·19-s + 0.732·23-s + 2.64·25-s − 0.192·27-s + 0.653·29-s − 1.42·31-s + 0.496·33-s + 0.956·37-s + 1.04·39-s − 0.574·41-s − 1.56·43-s + 0.636·45-s − 0.640·47-s − 0.132·51-s + 1.10·53-s − 1.64·55-s − 0.139·57-s + 0.801·59-s + 0.721·61-s − 3.45·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 21.3T + 125T^{2} \) |
| 11 | \( 1 + 31.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 363.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 343.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 69.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 468.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 563.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−8.309134339515903686182956295745, −7.15272227493015153272749073792, −6.76791638762154943905542383318, −5.54889687436081444478935091467, −5.39589298163870708551857900944, −4.61233679853267335025760416061, −2.94571380293260757707466393308, −2.29060285207794744930531709270, −1.33531859941937378928365961322, 0,
1.33531859941937378928365961322, 2.29060285207794744930531709270, 2.94571380293260757707466393308, 4.61233679853267335025760416061, 5.39589298163870708551857900944, 5.54889687436081444478935091467, 6.76791638762154943905542383318, 7.15272227493015153272749073792, 8.309134339515903686182956295745
