| L(s) = 1 | + 7-s + 3.46·11-s − 4·13-s + 6.92·17-s + 19-s + 3.46·23-s − 5·25-s − 3.46·29-s + 8·31-s + 2·37-s − 6.92·41-s + 8·43-s + 3.46·47-s + 49-s − 10.3·53-s − 13.8·59-s + 2·61-s + 2·67-s + 3.46·71-s + 14·73-s + 3.46·77-s + 14·79-s − 10.3·83-s + 13.8·89-s − 4·91-s + 8·97-s − 4·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.04·11-s − 1.10·13-s + 1.68·17-s + 0.229·19-s + 0.722·23-s − 25-s − 0.643·29-s + 1.43·31-s + 0.328·37-s − 1.08·41-s + 1.21·43-s + 0.505·47-s + 0.142·49-s − 1.42·53-s − 1.80·59-s + 0.256·61-s + 0.244·67-s + 0.411·71-s + 1.63·73-s + 0.394·77-s + 1.57·79-s − 1.14·83-s + 1.46·89-s − 0.419·91-s + 0.812·97-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224867115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224867115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.996994774262594005284166465632, −7.75073681995543965464618590410, −6.87037150575240516729161690830, −6.10860374961603052350653433191, −5.30678880024795367109817291558, −4.63328745194643210287311083983, −3.73239630914176293392467645183, −2.93600523775511834788550324634, −1.83887735495383727690659992435, −0.859489601820127144261081688810,
0.859489601820127144261081688810, 1.83887735495383727690659992435, 2.93600523775511834788550324634, 3.73239630914176293392467645183, 4.63328745194643210287311083983, 5.30678880024795367109817291558, 6.10860374961603052350653433191, 6.87037150575240516729161690830, 7.75073681995543965464618590410, 7.996994774262594005284166465632
