| L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s + 5·11-s − 6·13-s − 3·15-s − 5·17-s − 19-s − 21-s − 4·23-s + 4·25-s + 27-s + 6·29-s − 6·31-s + 5·33-s + 3·35-s − 8·37-s − 6·39-s − 8·41-s − 9·43-s − 3·45-s − 47-s − 6·49-s − 5·51-s + 2·53-s − 15·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 1.66·13-s − 0.774·15-s − 1.21·17-s − 0.229·19-s − 0.218·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.870·33-s + 0.507·35-s − 1.31·37-s − 0.960·39-s − 1.24·41-s − 1.37·43-s − 0.447·45-s − 0.145·47-s − 6/7·49-s − 0.700·51-s + 0.274·53-s − 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.585874925160168888904155563723, −8.743979403432894067002867605875, −8.098723763596542718825097439727, −6.98995551256525595204221630124, −6.70923923756293259323170265958, −4.96063409732475955643392332129, −4.09803140790119357857110127232, −3.38752685173358042429193565276, −2.02959224696734609090119252606, 0,
2.02959224696734609090119252606, 3.38752685173358042429193565276, 4.09803140790119357857110127232, 4.96063409732475955643392332129, 6.70923923756293259323170265958, 6.98995551256525595204221630124, 8.098723763596542718825097439727, 8.743979403432894067002867605875, 9.585874925160168888904155563723
