| L(s) = 1 | − 2·4-s − 0.837·7-s − 4.24·13-s + 4·16-s + 4.80·19-s − 5·25-s + 1.67·28-s − 10.9·31-s + 11.7·37-s − 0.551·43-s − 6.29·49-s + 8.49·52-s + 9.25·61-s − 8·64-s + 13.8·67-s + 17·73-s − 9.60·76-s + 17·79-s + 3.55·91-s − 8.41·97-s + 10·100-s + 17.5·103-s + 17·109-s − 3.34·112-s + ⋯ |
| L(s) = 1 | − 4-s − 0.316·7-s − 1.17·13-s + 16-s + 1.10·19-s − 25-s + 0.316·28-s − 1.96·31-s + 1.93·37-s − 0.0841·43-s − 0.899·49-s + 1.17·52-s + 1.18·61-s − 64-s + 1.69·67-s + 1.98·73-s − 1.10·76-s + 1.91·79-s + 0.373·91-s − 0.854·97-s + 100-s + 1.72·103-s + 1.62·109-s − 0.316·112-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
| \(L(1)\) |
\(\approx\) |
\(1.019410142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019410142\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 0.837T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 0.551T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 17T + 73T^{2} \) |
| 79 | \( 1 - 17T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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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
−9.106891607320551977526257864912, −7.967217051525851011082068016955, −7.60561750237883939091151616447, −6.58294048116123809598007654307, −5.52889909727005180405397970304, −5.06228575732628091462420949802, −4.05901671630168015299166167325, −3.32971937630583128763908622642, −2.13802991540737033238004293633, −0.62751342157998510005251033562,
0.62751342157998510005251033562, 2.13802991540737033238004293633, 3.32971937630583128763908622642, 4.05901671630168015299166167325, 5.06228575732628091462420949802, 5.52889909727005180405397970304, 6.58294048116123809598007654307, 7.60561750237883939091151616447, 7.967217051525851011082068016955, 9.106891607320551977526257864912
