L(s) = 1 | + 2.30·3-s − 3·5-s + 3.30·7-s + 2.30·9-s + 6.30·13-s − 6.90·15-s − 1.30·17-s + 2·19-s + 7.60·21-s + 1.30·23-s + 4·25-s − 1.60·27-s + 0.394·29-s − 0.605·31-s − 9.90·35-s + 7.60·37-s + 14.5·39-s + 8.21·41-s + 2.39·43-s − 6.90·45-s − 5.60·47-s + 3.90·49-s − 3·51-s + 2.60·53-s + 4.60·57-s − 13.8·59-s − 14.8·61-s + ⋯ |
L(s) = 1 | + 1.32·3-s − 1.34·5-s + 1.24·7-s + 0.767·9-s + 1.74·13-s − 1.78·15-s − 0.315·17-s + 0.458·19-s + 1.65·21-s + 0.271·23-s + 0.800·25-s − 0.308·27-s + 0.0732·29-s − 0.108·31-s − 1.67·35-s + 1.25·37-s + 2.32·39-s + 1.28·41-s + 0.365·43-s − 1.02·45-s − 0.817·47-s + 0.558·49-s − 0.420·51-s + 0.357·53-s + 0.610·57-s − 1.79·59-s − 1.89·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.204434830\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.204434830\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 + 0.605T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 + 5.60T + 47T^{2} \) |
| 53 | \( 1 - 2.60T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 6.21T + 67T^{2} \) |
| 71 | \( 1 + 6.90T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−10.78684176606250352780348090690, −9.220792238273381390054638834383, −8.671315156529340627781238131251, −7.82032309207390708230357363588, −7.65098058816185145766008554081, −6.10159033130350986165324583392, −4.59745577066120010943531232006, −3.87332638670442420979740841884, −2.94107401849814219392407337671, −1.41901172003878071098671570951,
1.41901172003878071098671570951, 2.94107401849814219392407337671, 3.87332638670442420979740841884, 4.59745577066120010943531232006, 6.10159033130350986165324583392, 7.65098058816185145766008554081, 7.82032309207390708230357363588, 8.671315156529340627781238131251, 9.220792238273381390054638834383, 10.78684176606250352780348090690