| L(s) = 1 | − 1.84·2-s + 1.41·4-s − 3.61·5-s + 3.84·7-s + 1.08·8-s + 6.67·10-s + 0.198·11-s − 0.883·13-s − 7.10·14-s − 4.82·16-s − 2.24·19-s − 5.10·20-s − 0.367·22-s − 2.55·23-s + 8.05·25-s + 1.63·26-s + 5.44·28-s + 2.24·29-s − 4.73·31-s + 6.75·32-s − 13.9·35-s + 6.98·37-s + 4.15·38-s − 3.91·40-s − 0.619·41-s − 7.17·43-s + 0.281·44-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.707·4-s − 1.61·5-s + 1.45·7-s + 0.382·8-s + 2.11·10-s + 0.0599·11-s − 0.245·13-s − 1.90·14-s − 1.20·16-s − 0.516·19-s − 1.14·20-s − 0.0783·22-s − 0.532·23-s + 1.61·25-s + 0.320·26-s + 1.02·28-s + 0.416·29-s − 0.851·31-s + 1.19·32-s − 2.34·35-s + 1.14·37-s + 0.674·38-s − 0.618·40-s − 0.0968·41-s − 1.09·43-s + 0.0424·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 0.198T + 11T^{2} \) |
| 13 | \( 1 + 0.883T + 13T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 6.98T + 37T^{2} \) |
| 41 | \( 1 + 0.619T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 - 5.18T + 53T^{2} \) |
| 59 | \( 1 + 5.01T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 0.281T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 9.51T + 83T^{2} \) |
| 89 | \( 1 + 4.33T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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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
−8.313165691595206548153517084475, −7.984574832665112577823617252401, −7.42742567132790902446767106040, −6.63318166621970144752967909239, −5.15561265256009784879750374089, −4.47119117351333904763390870602, −3.76492916021183318594812177182, −2.28428061267952388849545711370, −1.19062577925597326328792368256, 0,
1.19062577925597326328792368256, 2.28428061267952388849545711370, 3.76492916021183318594812177182, 4.47119117351333904763390870602, 5.15561265256009784879750374089, 6.63318166621970144752967909239, 7.42742567132790902446767106040, 7.984574832665112577823617252401, 8.313165691595206548153517084475
