| L(s) = 1 | − 1.39·2-s − 0.0647·4-s − 4.19·5-s − 0.214·7-s + 2.87·8-s + 5.83·10-s + 2.45·11-s + 1.29·13-s + 0.298·14-s − 3.86·16-s − 3.20·17-s − 6.32·19-s + 0.271·20-s − 3.41·22-s − 7.90·23-s + 12.6·25-s − 1.80·26-s + 0.0139·28-s + 5.02·29-s − 4.72·31-s − 0.366·32-s + 4.45·34-s + 0.902·35-s − 0.0218·37-s + 8.79·38-s − 12.0·40-s + 6.28·41-s + ⋯ |
| L(s) = 1 | − 0.983·2-s − 0.0323·4-s − 1.87·5-s − 0.0812·7-s + 1.01·8-s + 1.84·10-s + 0.740·11-s + 0.359·13-s + 0.0798·14-s − 0.966·16-s − 0.776·17-s − 1.45·19-s + 0.0608·20-s − 0.728·22-s − 1.64·23-s + 2.52·25-s − 0.353·26-s + 0.00263·28-s + 0.933·29-s − 0.849·31-s − 0.0647·32-s + 0.764·34-s + 0.152·35-s − 0.00358·37-s + 1.42·38-s − 1.90·40-s + 0.981·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3216354316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3216354316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 7 | \( 1 + 0.214T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 + 0.0218T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 9.93T + 53T^{2} \) |
| 59 | \( 1 - 6.22T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 1.89T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 3.61T + 83T^{2} \) |
| 89 | \( 1 - 0.669T + 89T^{2} \) |
| 97 | \( 1 - 6.62T + 97T^{2} \) |
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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
−8.700321882829435766978454149122, −8.464030372397085460652669620679, −7.81886448596875074822493773921, −6.98166381319167763515208636172, −6.29531184440331933346537838625, −4.66627269792390222485189553061, −4.21586724351706443137673864131, −3.46546406667097950834029538559, −1.85782642873151336926071146308, −0.42801059137348282852090835010,
0.42801059137348282852090835010, 1.85782642873151336926071146308, 3.46546406667097950834029538559, 4.21586724351706443137673864131, 4.66627269792390222485189553061, 6.29531184440331933346537838625, 6.98166381319167763515208636172, 7.81886448596875074822493773921, 8.464030372397085460652669620679, 8.700321882829435766978454149122
