| L(s) = 1 | − 3-s + 9-s + 4·11-s − 13-s + 6·17-s − 8·19-s − 4·23-s − 27-s − 2·29-s − 4·31-s − 4·33-s − 10·37-s + 39-s − 6·41-s − 8·43-s + 8·47-s − 6·51-s + 2·53-s + 8·57-s + 8·59-s + 2·61-s + 4·67-s + 4·69-s + 8·71-s − 10·73-s − 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 1.45·17-s − 1.83·19-s − 0.834·23-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s − 1.64·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s + 1.16·47-s − 0.840·51-s + 0.274·53-s + 1.05·57-s + 1.04·59-s + 0.256·61-s + 0.488·67-s + 0.481·69-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.53842670368392, −12.18214179557714, −11.85360924476044, −11.48527097008510, −10.85918333661579, −10.35824682079105, −10.14146969914435, −9.643175267954344, −9.020432173618637, −8.625560451687640, −8.229772735894968, −7.598134071874518, −7.017155494153730, −6.793604237984310, −6.139677480750668, −5.846223723847065, −5.218071000533785, −4.849382463422822, −4.094734303835775, −3.740879013373586, −3.412936325091368, −2.458410497520605, −1.841561214353428, −1.511447758103980, −0.6660397021677337, 0,
0.6660397021677337, 1.511447758103980, 1.841561214353428, 2.458410497520605, 3.412936325091368, 3.740879013373586, 4.094734303835775, 4.849382463422822, 5.218071000533785, 5.846223723847065, 6.139677480750668, 6.793604237984310, 7.017155494153730, 7.598134071874518, 8.229772735894968, 8.625560451687640, 9.020432173618637, 9.643175267954344, 10.14146969914435, 10.35824682079105, 10.85918333661579, 11.48527097008510, 11.85360924476044, 12.18214179557714, 12.53842670368392
