| L(s) = 1 | − 2-s + 4-s − 4.24·5-s − 8-s + 4.24·10-s + 11-s + 16-s + 5.65·17-s − 4.24·20-s − 22-s − 6·23-s + 12.9·25-s − 2·29-s − 1.41·31-s − 32-s − 5.65·34-s − 10·37-s + 4.24·40-s + 11.3·41-s − 8·43-s + 44-s + 6·46-s − 4.24·47-s − 12.9·50-s − 8·53-s − 4.24·55-s + 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.89·5-s − 0.353·8-s + 1.34·10-s + 0.301·11-s + 0.250·16-s + 1.37·17-s − 0.948·20-s − 0.213·22-s − 1.25·23-s + 2.59·25-s − 0.371·29-s − 0.254·31-s − 0.176·32-s − 0.970·34-s − 1.64·37-s + 0.670·40-s + 1.76·41-s − 1.21·43-s + 0.150·44-s + 0.884·46-s − 0.618·47-s − 1.83·50-s − 1.09·53-s − 0.572·55-s + 0.262·58-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 | \( 1 + 4.24T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−7.53920741335489587690917938226, −6.93173826762667047360434676102, −6.17381493138969563471037964906, −5.23903062470301943226404285557, −4.44841502794501465897132187689, −3.48398593488093022778896486393, −3.39699600197380146673885232977, −2.01619476394609927902420789822, −0.925726703647732877729297116843, 0,
0.925726703647732877729297116843, 2.01619476394609927902420789822, 3.39699600197380146673885232977, 3.48398593488093022778896486393, 4.44841502794501465897132187689, 5.23903062470301943226404285557, 6.17381493138969563471037964906, 6.93173826762667047360434676102, 7.53920741335489587690917938226
