| L(s) = 1 | − 2-s + 4-s + 1.41·5-s − 8-s − 1.41·10-s + 11-s + 5.65·13-s + 16-s − 1.41·17-s − 4.24·19-s + 1.41·20-s − 22-s + 4·23-s − 2.99·25-s − 5.65·26-s − 4.24·31-s − 32-s + 1.41·34-s − 6·37-s + 4.24·38-s − 1.41·40-s − 4.24·41-s − 4·43-s + 44-s − 4·46-s + 1.41·47-s + 2.99·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.632·5-s − 0.353·8-s − 0.447·10-s + 0.301·11-s + 1.56·13-s + 0.250·16-s − 0.342·17-s − 0.973·19-s + 0.316·20-s − 0.213·22-s + 0.834·23-s − 0.599·25-s − 1.10·26-s − 0.762·31-s − 0.176·32-s + 0.242·34-s − 0.986·37-s + 0.688·38-s − 0.223·40-s − 0.662·41-s − 0.609·43-s + 0.150·44-s − 0.589·46-s + 0.206·47-s + 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−7.26721254272450337064002970402, −6.70420178923516279929851793442, −6.06073444193383960591800673440, −5.56407894486452325254738461243, −4.52289302762779595756655046820, −3.70150398585820719141554591090, −2.93805677255495409999842716933, −1.84075009332519664123753407761, −1.38208657977308203308057694588, 0,
1.38208657977308203308057694588, 1.84075009332519664123753407761, 2.93805677255495409999842716933, 3.70150398585820719141554591090, 4.52289302762779595756655046820, 5.56407894486452325254738461243, 6.06073444193383960591800673440, 6.70420178923516279929851793442, 7.26721254272450337064002970402
