L(s) = 1 | − 3-s − 5-s + 5·7-s − 2·9-s − 5·11-s + 15-s − 17-s − 3·19-s − 5·21-s + 3·23-s + 25-s + 5·27-s − 29-s + 5·33-s − 5·35-s + 7·37-s − 5·41-s + 5·43-s + 2·45-s + 12·47-s + 18·49-s + 51-s + 2·53-s + 5·55-s + 3·57-s − 11·59-s − 13·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.88·7-s − 2/3·9-s − 1.50·11-s + 0.258·15-s − 0.242·17-s − 0.688·19-s − 1.09·21-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s + 0.870·33-s − 0.845·35-s + 1.15·37-s − 0.780·41-s + 0.762·43-s + 0.298·45-s + 1.75·47-s + 18/7·49-s + 0.140·51-s + 0.274·53-s + 0.674·55-s + 0.397·57-s − 1.43·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.282987876\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.282987876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−8.462549198830603355529116791466, −7.84679795274503102057999312076, −7.40339786952471178809683886164, −6.20792994116419462452057948108, −5.40946224975916314391744621508, −4.89316421372881460824628509013, −4.25007576051372465263386573804, −2.87406148538882991766898136086, −2.04664563282205179191477220269, −0.68333426554367118099719718411,
0.68333426554367118099719718411, 2.04664563282205179191477220269, 2.87406148538882991766898136086, 4.25007576051372465263386573804, 4.89316421372881460824628509013, 5.40946224975916314391744621508, 6.20792994116419462452057948108, 7.40339786952471178809683886164, 7.84679795274503102057999312076, 8.462549198830603355529116791466