| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 4·13-s + 14-s + 16-s + 2·17-s − 8·19-s − 8·23-s + 4·26-s − 28-s + 8·29-s + 4·31-s − 32-s − 2·34-s + 8·37-s + 8·38-s + 12·41-s − 8·43-s + 8·46-s + 4·47-s + 49-s − 4·52-s + 6·53-s + 56-s − 8·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 1.66·23-s + 0.784·26-s − 0.188·28-s + 1.48·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s + 1.31·37-s + 1.29·38-s + 1.87·41-s − 1.21·43-s + 1.17·46-s + 0.583·47-s + 1/7·49-s − 0.554·52-s + 0.824·53-s + 0.133·56-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9218233322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9218233322\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.614349031704417557447662412358, −8.017961683457142854769185438055, −7.34086440986198482728227467528, −6.38070520827225220517753534606, −6.00064548691327187706002474290, −4.73599706324926864372938537137, −4.01305040865091997480415377012, −2.73906382214215656416049101671, −2.11417522475526314640919317579, −0.61807987101016103476204337955,
0.61807987101016103476204337955, 2.11417522475526314640919317579, 2.73906382214215656416049101671, 4.01305040865091997480415377012, 4.73599706324926864372938537137, 6.00064548691327187706002474290, 6.38070520827225220517753534606, 7.34086440986198482728227467528, 8.017961683457142854769185438055, 8.614349031704417557447662412358
