| L(s) = 1 | − 3·3-s − 3.90·5-s − 36.4·7-s + 9·9-s − 19.1·11-s + 13·13-s + 11.7·15-s − 83.8·17-s − 46.8·19-s + 109.·21-s − 103.·23-s − 109.·25-s − 27·27-s + 108.·29-s + 147.·31-s + 57.5·33-s + 142.·35-s − 160.·37-s − 39·39-s + 231.·41-s + 340.·43-s − 35.1·45-s − 119.·47-s + 982.·49-s + 251.·51-s − 732.·53-s + 75.0·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.349·5-s − 1.96·7-s + 0.333·9-s − 0.526·11-s + 0.277·13-s + 0.201·15-s − 1.19·17-s − 0.565·19-s + 1.13·21-s − 0.941·23-s − 0.877·25-s − 0.192·27-s + 0.693·29-s + 0.854·31-s + 0.303·33-s + 0.687·35-s − 0.710·37-s − 0.160·39-s + 0.881·41-s + 1.20·43-s − 0.116·45-s − 0.371·47-s + 2.86·49-s + 0.690·51-s − 1.89·53-s + 0.183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4771513087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4771513087\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 3.90T + 125T^{2} \) |
| 7 | \( 1 + 36.4T + 343T^{2} \) |
| 11 | \( 1 + 19.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 732.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 10.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 858.T + 9.12e5T^{2} \) |
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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
−10.19878462038448519033167572878, −9.518607595913421051732423588578, −8.524297692716032559794292086279, −7.38411910005463762053578377412, −6.38997823609475309454584224457, −6.01378221147016721218867316155, −4.53299357971379930481855150297, −3.59326293987790616391690387734, −2.41046654086293974831104480994, −0.39328445178208225741188991741,
0.39328445178208225741188991741, 2.41046654086293974831104480994, 3.59326293987790616391690387734, 4.53299357971379930481855150297, 6.01378221147016721218867316155, 6.38997823609475309454584224457, 7.38411910005463762053578377412, 8.524297692716032559794292086279, 9.518607595913421051732423588578, 10.19878462038448519033167572878
