| L(s) = 1 | + 2-s + 4-s + 8-s − 4.24·11-s − 2.24·13-s + 16-s − 2.24·19-s − 4.24·22-s − 1.24·23-s − 5·25-s − 2.24·26-s + 4.24·29-s + 9.24·31-s + 32-s − 4·37-s − 2.24·38-s + 11.4·41-s + 10.4·43-s − 4.24·44-s − 1.24·46-s + 4.75·47-s − 5·50-s − 2.24·52-s − 4.24·53-s + 4.24·58-s − 2.24·61-s + 9.24·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s − 1.27·11-s − 0.621·13-s + 0.250·16-s − 0.514·19-s − 0.904·22-s − 0.259·23-s − 25-s − 0.439·26-s + 0.787·29-s + 1.66·31-s + 0.176·32-s − 0.657·37-s − 0.363·38-s + 1.79·41-s + 1.59·43-s − 0.639·44-s − 0.183·46-s + 0.693·47-s − 0.707·50-s − 0.310·52-s − 0.582·53-s + 0.557·58-s − 0.287·61-s + 1.17·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.672036898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.672036898\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 0.242T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−7.70161476514405457012856076169, −7.24369061841218580801201993808, −6.14816792832787184383632766563, −5.89066765527308498626847438714, −4.83137922198848655727291256855, −4.52313541151255672851331147525, −3.53330106696162525732118237625, −2.60144274372572179159405572247, −2.18301619449332407510495761933, −0.69887583669390081128176484699,
0.69887583669390081128176484699, 2.18301619449332407510495761933, 2.60144274372572179159405572247, 3.53330106696162525732118237625, 4.52313541151255672851331147525, 4.83137922198848655727291256855, 5.89066765527308498626847438714, 6.14816792832787184383632766563, 7.24369061841218580801201993808, 7.70161476514405457012856076169
