| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 11-s + 13-s − 15-s + 6·17-s + 4·19-s + 2·21-s + 25-s − 27-s − 2·31-s − 33-s − 2·35-s − 4·37-s − 39-s + 6·41-s − 8·43-s + 45-s − 3·49-s − 6·51-s + 12·53-s + 55-s − 4·57-s − 10·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.359·31-s − 0.174·33-s − 0.338·35-s − 0.657·37-s − 0.160·39-s + 0.937·41-s − 1.21·43-s + 0.149·45-s − 3/7·49-s − 0.840·51-s + 1.64·53-s + 0.134·55-s − 0.529·57-s − 1.28·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.20499133638491, −14.70734461641693, −14.16332876603166, −13.54379309814744, −13.20889357038713, −12.52344569089070, −12.02866961274762, −11.72371810136727, −10.93046815711762, −10.38635613684128, −9.920448577581317, −9.487276692550654, −8.924175393706358, −8.205042018695622, −7.455549657706256, −7.061081281044068, −6.357196428205980, −5.799041575752530, −5.449074341887939, −4.733211950764393, −3.883707360791518, −3.308308702978421, −2.724261915202303, −1.594752769087487, −1.075648896019044, 0,
1.075648896019044, 1.594752769087487, 2.724261915202303, 3.308308702978421, 3.883707360791518, 4.733211950764393, 5.449074341887939, 5.799041575752530, 6.357196428205980, 7.061081281044068, 7.455549657706256, 8.205042018695622, 8.924175393706358, 9.487276692550654, 9.920448577581317, 10.38635613684128, 10.93046815711762, 11.72371810136727, 12.02866961274762, 12.52344569089070, 13.20889357038713, 13.54379309814744, 14.16332876603166, 14.70734461641693, 15.20499133638491
