L(s) = 1 | + 0.414·3-s − 4.82·7-s − 2.82·9-s + 3.24·11-s + 5.65·13-s − 5.82·17-s − 4.41·19-s − 1.99·21-s + 0.828·23-s − 2.41·27-s + 8·29-s + 4.82·31-s + 1.34·33-s + 3.65·37-s + 2.34·39-s − 0.656·41-s − 10·43-s + 1.65·47-s + 16.3·49-s − 2.41·51-s − 3.65·53-s − 1.82·57-s − 7.65·59-s + 6·61-s + 13.6·63-s + 11.2·67-s + 0.343·69-s + ⋯ |
L(s) = 1 | + 0.239·3-s − 1.82·7-s − 0.942·9-s + 0.977·11-s + 1.56·13-s − 1.41·17-s − 1.01·19-s − 0.436·21-s + 0.172·23-s − 0.464·27-s + 1.48·29-s + 0.867·31-s + 0.233·33-s + 0.601·37-s + 0.375·39-s − 0.102·41-s − 1.52·43-s + 0.241·47-s + 2.33·49-s − 0.338·51-s − 0.502·53-s − 0.242·57-s − 0.996·59-s + 0.768·61-s + 1.72·63-s + 1.37·67-s + 0.0413·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.357124136\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357124136\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 0.656T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 - 0.171T + 73T^{2} \) |
| 79 | \( 1 - 7.17T + 79T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−8.734754502305828996519533703056, −8.240564557181795698856988048508, −6.77796773614085822495237977472, −6.36049509031105308230413033312, −6.11062069679473812150628552125, −4.66859282256240915804030474104, −3.73687536721649377810951254013, −3.19129636462519200281307447056, −2.22681444461870833576452679132, −0.66951563492797050801146136874,
0.66951563492797050801146136874, 2.22681444461870833576452679132, 3.19129636462519200281307447056, 3.73687536721649377810951254013, 4.66859282256240915804030474104, 6.11062069679473812150628552125, 6.36049509031105308230413033312, 6.77796773614085822495237977472, 8.240564557181795698856988048508, 8.734754502305828996519533703056