| L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s + 2·11-s − 4·15-s + 17-s + 2·21-s + 11·25-s − 27-s − 6·29-s − 2·31-s − 2·33-s − 8·35-s − 4·37-s − 8·41-s + 4·43-s + 4·45-s + 4·47-s − 3·49-s − 51-s + 2·53-s + 8·55-s − 4·59-s + 6·61-s − 2·63-s + 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.03·15-s + 0.242·17-s + 0.436·21-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.348·33-s − 1.35·35-s − 0.657·37-s − 1.24·41-s + 0.609·43-s + 0.596·45-s + 0.583·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s + 1.07·55-s − 0.520·59-s + 0.768·61-s − 0.251·63-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.526039707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.526039707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.94248784423642, −13.68650147327312, −13.16418105586589, −12.69677781215780, −12.28568555018833, −11.70325843533296, −10.90770906793904, −10.69387102381293, −9.953602254257388, −9.652060925874147, −9.290122935066501, −8.743616821308913, −8.022326683577708, −7.124360247404590, −6.743200598067114, −6.361269676457194, −5.633282294836081, −5.501601350483808, −4.820826633328422, −3.953644485530573, −3.388939155065432, −2.621557927821585, −1.916811805396462, −1.439386444258778, −0.5460074053138270,
0.5460074053138270, 1.439386444258778, 1.916811805396462, 2.621557927821585, 3.388939155065432, 3.953644485530573, 4.820826633328422, 5.501601350483808, 5.633282294836081, 6.361269676457194, 6.743200598067114, 7.124360247404590, 8.022326683577708, 8.743616821308913, 9.290122935066501, 9.652060925874147, 9.953602254257388, 10.69387102381293, 10.90770906793904, 11.70325843533296, 12.28568555018833, 12.69677781215780, 13.16418105586589, 13.68650147327312, 13.94248784423642
