| L(s) = 1 | − 6.78e10·2-s + 3.41e21·4-s − 3.78e29·7-s − 1.51e32·8-s + 2.50e33·9-s − 1.35e36·11-s + 2.56e40·14-s + 6.25e42·16-s − 1.69e44·18-s + 9.17e46·22-s − 9.13e47·23-s + 8.47e48·25-s − 1.29e51·28-s − 2.66e51·29-s − 2.44e53·32-s + 8.55e54·36-s + 3.86e53·37-s − 2.07e57·43-s − 4.62e57·44-s + 6.19e58·46-s + 1.43e59·49-s − 5.74e59·50-s + 4.10e60·53-s + 5.74e61·56-s + 1.80e62·58-s − 9.48e62·63-s + 9.21e63·64-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.89·4-s − 7-s − 3.73·8-s + 9-s − 0.481·11-s + 1.97·14-s + 4.48·16-s − 1.97·18-s + 0.950·22-s − 1.99·23-s + 25-s − 2.89·28-s − 1.74·29-s − 5.11·32-s + 2.89·36-s + 0.0500·37-s − 1.39·43-s − 1.39·44-s + 3.93·46-s + 49-s − 1.97·50-s + 1.83·53-s + 3.73·56-s + 3.44·58-s − 63-s + 5.60·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{71}{2})\) |
\(\approx\) |
\(0.3689663112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3689663112\) |
| \(L(36)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{35} T \) |
| good | 2 | \( 1 + 67809066807 T + p^{70} T^{2} \) |
| 3 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 5 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 11 | \( 1 + \)\(13\!\cdots\!26\)\( T + p^{70} T^{2} \) |
| 13 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 17 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 19 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 23 | \( 1 + \)\(91\!\cdots\!18\)\( T + p^{70} T^{2} \) |
| 29 | \( 1 + \)\(26\!\cdots\!74\)\( T + p^{70} T^{2} \) |
| 31 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 37 | \( 1 - \)\(38\!\cdots\!18\)\( T + p^{70} T^{2} \) |
| 41 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 43 | \( 1 + \)\(20\!\cdots\!18\)\( T + p^{70} T^{2} \) |
| 47 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 53 | \( 1 - \)\(41\!\cdots\!86\)\( T + p^{70} T^{2} \) |
| 59 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 61 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 67 | \( 1 - \)\(17\!\cdots\!18\)\( T + p^{70} T^{2} \) |
| 71 | \( 1 + \)\(11\!\cdots\!26\)\( T + p^{70} T^{2} \) |
| 73 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 79 | \( 1 + \)\(15\!\cdots\!74\)\( T + p^{70} T^{2} \) |
| 83 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 89 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
| 97 | \( ( 1 - p^{35} T )( 1 + p^{35} T ) \) |
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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
−10.40552969301470889902877361920, −9.902886845214117516075865465897, −8.872982972826651648018077245922, −7.68349808107552348336220795114, −6.88838438474619390677790377384, −5.85796995560974791760867512757, −3.63535676047594951022701395154, −2.41274370831788822384743295267, −1.49956057662175425489848521357, −0.34066893502905269864169443766,
0.34066893502905269864169443766, 1.49956057662175425489848521357, 2.41274370831788822384743295267, 3.63535676047594951022701395154, 5.85796995560974791760867512757, 6.88838438474619390677790377384, 7.68349808107552348336220795114, 8.872982972826651648018077245922, 9.902886845214117516075865465897, 10.40552969301470889902877361920
