L(s) = 1 | + 0.826·3-s − 1.37·5-s − 4.67·7-s − 2.31·9-s − 4.79·11-s − 1.48·13-s − 1.13·15-s − 3.39·17-s + 6.13·19-s − 3.86·21-s + 0.540·23-s − 3.10·25-s − 4.39·27-s − 2.60·29-s + 4.02·31-s − 3.96·33-s + 6.44·35-s + 8.19·37-s − 1.22·39-s + 2.41·41-s − 9.14·43-s + 3.19·45-s + 4.35·47-s + 14.8·49-s − 2.80·51-s − 11.0·53-s + 6.60·55-s + ⋯ |
L(s) = 1 | + 0.477·3-s − 0.616·5-s − 1.76·7-s − 0.772·9-s − 1.44·11-s − 0.410·13-s − 0.294·15-s − 0.823·17-s + 1.40·19-s − 0.843·21-s + 0.112·23-s − 0.620·25-s − 0.845·27-s − 0.483·29-s + 0.723·31-s − 0.689·33-s + 1.08·35-s + 1.34·37-s − 0.196·39-s + 0.376·41-s − 1.39·43-s + 0.475·45-s + 0.635·47-s + 2.12·49-s − 0.393·51-s − 1.51·53-s + 0.891·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5749717543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5749717543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 0.826T + 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 - 0.540T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 4.02T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 0.771T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 8.84T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 - 1.72T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.381961433058007850616707712946, −7.70457651699823103914550035808, −7.16204537852634925096425910306, −6.19338793614610134480989333951, −5.58641939267047963190992283488, −4.62578721249282501175916928249, −3.53913968163668691446976715338, −2.99940263595589543030212752542, −2.38951042682439163398560662565, −0.38816784900192442095382127137,
0.38816784900192442095382127137, 2.38951042682439163398560662565, 2.99940263595589543030212752542, 3.53913968163668691446976715338, 4.62578721249282501175916928249, 5.58641939267047963190992283488, 6.19338793614610134480989333951, 7.16204537852634925096425910306, 7.70457651699823103914550035808, 8.381961433058007850616707712946