| L(s) = 1 | − 49·7-s + 81·9-s − 206·11-s + 734·23-s + 625·25-s − 1.23e3·29-s + 1.29e3·37-s − 334·43-s + 2.40e3·49-s + 5.58e3·53-s − 3.96e3·63-s + 4.94e3·67-s − 2.91e3·71-s + 1.00e4·77-s + 3.64e3·79-s + 6.56e3·81-s − 1.66e4·99-s + 1.16e4·107-s + 1.25e4·109-s + 2.37e4·113-s + ⋯ |
| L(s) = 1 | − 7-s + 9-s − 1.70·11-s + 1.38·23-s + 25-s − 1.46·29-s + 0.945·37-s − 0.180·43-s + 49-s + 1.98·53-s − 63-s + 1.10·67-s − 0.578·71-s + 1.70·77-s + 0.584·79-s + 81-s − 1.70·99-s + 1.02·107-s + 1.05·109-s + 1.85·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.576279695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576279695\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 + 206 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 - 734 T + p^{4} T^{2} \) |
| 29 | \( 1 + 1234 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 - 1294 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 334 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 - 5582 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 - 4946 T + p^{4} T^{2} \) |
| 71 | \( 1 + 2914 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 - 3646 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−10.40400159871017845817282482464, −9.713555352646143021945384394052, −8.761704619550267261495116396398, −7.53174631305035294399807229535, −6.94672474132811524374364967916, −5.70528087362642943563972725868, −4.74674742124166984954658261881, −3.44217921918032755984935478283, −2.39148155442852206233792784718, −0.69647545293277281336935830312,
0.69647545293277281336935830312, 2.39148155442852206233792784718, 3.44217921918032755984935478283, 4.74674742124166984954658261881, 5.70528087362642943563972725868, 6.94672474132811524374364967916, 7.53174631305035294399807229535, 8.761704619550267261495116396398, 9.713555352646143021945384394052, 10.40400159871017845817282482464
