| L(s) = 1 | + 2·3-s − 2·4-s − 3·5-s + 9-s + 3·11-s − 4·12-s + 4·13-s − 6·15-s + 4·16-s + 3·17-s − 19-s + 6·20-s + 4·25-s − 4·27-s + 6·29-s + 4·31-s + 6·33-s − 2·36-s + 2·37-s + 8·39-s + 6·41-s − 43-s − 6·44-s − 3·45-s + 3·47-s + 8·48-s + 6·51-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.34·5-s + 1/3·9-s + 0.904·11-s − 1.15·12-s + 1.10·13-s − 1.54·15-s + 16-s + 0.727·17-s − 0.229·19-s + 1.34·20-s + 4/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.04·33-s − 1/3·36-s + 0.328·37-s + 1.28·39-s + 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.447·45-s + 0.437·47-s + 1.15·48-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559894300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559894300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.773454048311221733670015344255, −9.016326316893545637066132714015, −8.263027957978573626933432932127, −8.077715259365695641127152270117, −6.88766610803394496528527618774, −5.63748710284752228673506815622, −4.18976040195982393758597830087, −3.89473302068219023186066736123, −2.94620381926703000046884743581, −0.997604860460902876433339628282,
0.997604860460902876433339628282, 2.94620381926703000046884743581, 3.89473302068219023186066736123, 4.18976040195982393758597830087, 5.63748710284752228673506815622, 6.88766610803394496528527618774, 8.077715259365695641127152270117, 8.263027957978573626933432932127, 9.016326316893545637066132714015, 9.773454048311221733670015344255
