| L(s) = 1 | − 9·3-s − 92.8·5-s + 81·9-s + 140.·11-s − 1.11e3·13-s + 835.·15-s + 54.8·17-s − 1.71e3·19-s − 3.28e3·23-s + 5.49e3·25-s − 729·27-s − 3.79e3·29-s − 4.84e3·31-s − 1.26e3·33-s − 1.13e4·37-s + 1.00e4·39-s − 1.03e4·41-s + 7.13e3·43-s − 7.51e3·45-s − 1.64e4·47-s − 493.·51-s − 2.09e4·53-s − 1.30e4·55-s + 1.54e4·57-s − 3.62e4·59-s + 4.94e3·61-s + 1.03e5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.66·5-s + 0.333·9-s + 0.350·11-s − 1.82·13-s + 0.958·15-s + 0.0460·17-s − 1.08·19-s − 1.29·23-s + 1.75·25-s − 0.192·27-s − 0.837·29-s − 0.905·31-s − 0.202·33-s − 1.36·37-s + 1.05·39-s − 0.964·41-s + 0.588·43-s − 0.553·45-s − 1.08·47-s − 0.0265·51-s − 1.02·53-s − 0.582·55-s + 0.628·57-s − 1.35·59-s + 0.170·61-s + 3.02·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.007294086385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007294086385\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 92.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 140.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 54.8T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.64e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.62e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.94e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.56e4T + 8.58e9T^{2} \) |
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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
−10.06410461390824399159984877283, −8.991866728778416579763455753124, −7.905174224662467604886480555458, −7.36357438026478575416290478608, −6.44844864753482093055307576236, −5.10220746236652971520742976048, −4.32774520428248031298180763401, −3.44234435551543409629463844544, −1.91963735283839528513543943716, −0.04074361905608839665991496910,
0.04074361905608839665991496910, 1.91963735283839528513543943716, 3.44234435551543409629463844544, 4.32774520428248031298180763401, 5.10220746236652971520742976048, 6.44844864753482093055307576236, 7.36357438026478575416290478608, 7.905174224662467604886480555458, 8.991866728778416579763455753124, 10.06410461390824399159984877283
