| L(s) = 1 | − 5-s + 7-s + 11-s − 2·17-s − 23-s + 25-s − 3·29-s + 11·31-s − 35-s + 2·37-s − 9·41-s − 43-s + 7·47-s − 6·49-s − 3·53-s − 55-s + 4·59-s − 8·61-s − 8·67-s − 10·71-s − 7·73-s + 77-s + 14·79-s + 2·85-s + 12·89-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.301·11-s − 0.485·17-s − 0.208·23-s + 1/5·25-s − 0.557·29-s + 1.97·31-s − 0.169·35-s + 0.328·37-s − 1.40·41-s − 0.152·43-s + 1.02·47-s − 6/7·49-s − 0.412·53-s − 0.134·55-s + 0.520·59-s − 1.02·61-s − 0.977·67-s − 1.18·71-s − 0.819·73-s + 0.113·77-s + 1.57·79-s + 0.216·85-s + 1.27·89-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780505499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780505499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.42734742147182, −12.02332405100309, −11.76474092955828, −11.30839986728891, −10.77261485152934, −10.36184768034318, −9.920059174006807, −9.340140329605121, −8.876563104058116, −8.467717375845989, −7.958425106602131, −7.602597780135146, −7.048817063238586, −6.454351307907570, −6.208528326722990, −5.515327748643613, −4.920204385932776, −4.494413812593775, −4.137587262023116, −3.394704617127272, −3.008838237116424, −2.308482859844429, −1.717198731106940, −1.113139341415994, −0.3715411127756459,
0.3715411127756459, 1.113139341415994, 1.717198731106940, 2.308482859844429, 3.008838237116424, 3.394704617127272, 4.137587262023116, 4.494413812593775, 4.920204385932776, 5.515327748643613, 6.208528326722990, 6.454351307907570, 7.048817063238586, 7.602597780135146, 7.958425106602131, 8.467717375845989, 8.876563104058116, 9.340140329605121, 9.920059174006807, 10.36184768034318, 10.77261485152934, 11.30839986728891, 11.76474092955828, 12.02332405100309, 12.42734742147182
