| L(s) = 1 | + 2-s + 3-s + 4-s − 3.20·5-s + 6-s + 1.51·7-s + 8-s + 9-s − 3.20·10-s − 6.53·11-s + 12-s + 2.88·13-s + 1.51·14-s − 3.20·15-s + 16-s − 1.19·17-s + 18-s + 0.555·19-s − 3.20·20-s + 1.51·21-s − 6.53·22-s + 24-s + 5.26·25-s + 2.88·26-s + 27-s + 1.51·28-s − 8.66·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.43·5-s + 0.408·6-s + 0.571·7-s + 0.353·8-s + 0.333·9-s − 1.01·10-s − 1.97·11-s + 0.288·12-s + 0.800·13-s + 0.404·14-s − 0.827·15-s + 0.250·16-s − 0.290·17-s + 0.235·18-s + 0.127·19-s − 0.716·20-s + 0.330·21-s − 1.39·22-s + 0.204·24-s + 1.05·25-s + 0.566·26-s + 0.192·27-s + 0.285·28-s − 1.60·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 + 6.53T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 - 0.555T + 19T^{2} \) |
| 29 | \( 1 + 8.66T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 + 1.45T + 37T^{2} \) |
| 41 | \( 1 + 1.84T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 3.96T + 59T^{2} \) |
| 61 | \( 1 - 9.03T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 + 9.45T + 71T^{2} \) |
| 73 | \( 1 + 0.627T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−8.293190842870513093471130608191, −7.41437140859080479144146324708, −7.22645972756151057246600551387, −5.77186202574109996718185909413, −5.12634151932687479363154870898, −4.26469116913936796843321105273, −3.59800487813110209066633600364, −2.83924335984516125066128127720, −1.76511663006087009650902968967, 0,
1.76511663006087009650902968967, 2.83924335984516125066128127720, 3.59800487813110209066633600364, 4.26469116913936796843321105273, 5.12634151932687479363154870898, 5.77186202574109996718185909413, 7.22645972756151057246600551387, 7.41437140859080479144146324708, 8.293190842870513093471130608191
