| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 6·13-s + 2·15-s − 2·17-s − 4·19-s − 21-s + 4·23-s − 25-s + 27-s − 10·29-s + 8·31-s − 2·35-s + 6·37-s + 6·39-s − 2·41-s + 4·43-s + 2·45-s − 8·47-s + 49-s − 2·51-s − 10·53-s − 4·57-s − 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.338·35-s + 0.986·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797346058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797346058\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−11.36300048410957114627275585967, −10.59545160919399150353614339771, −9.530718402063992031149017201598, −8.905190686502749891901965349669, −7.917190719920927542416951015280, −6.54428519689900750892915035611, −5.90015243373728427076887013140, −4.35622436888448856514855173618, −3.10934300001651305626809863177, −1.70572377211895013061556457420,
1.70572377211895013061556457420, 3.10934300001651305626809863177, 4.35622436888448856514855173618, 5.90015243373728427076887013140, 6.54428519689900750892915035611, 7.917190719920927542416951015280, 8.905190686502749891901965349669, 9.530718402063992031149017201598, 10.59545160919399150353614339771, 11.36300048410957114627275585967
