| L(s) = 1 | + 2-s − 1.28·3-s + 4-s − 1.35·5-s − 1.28·6-s − 3.16·7-s + 8-s − 1.35·9-s − 1.35·10-s − 1.28·12-s − 0.907·13-s − 3.16·14-s + 1.74·15-s + 16-s − 2.83·17-s − 1.35·18-s + 19-s − 1.35·20-s + 4.06·21-s + 2.86·23-s − 1.28·24-s − 3.15·25-s − 0.907·26-s + 5.58·27-s − 3.16·28-s + 4.97·29-s + 1.74·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.741·3-s + 0.5·4-s − 0.607·5-s − 0.524·6-s − 1.19·7-s + 0.353·8-s − 0.450·9-s − 0.429·10-s − 0.370·12-s − 0.251·13-s − 0.846·14-s + 0.450·15-s + 0.250·16-s − 0.687·17-s − 0.318·18-s + 0.229·19-s − 0.303·20-s + 0.887·21-s + 0.596·23-s − 0.262·24-s − 0.630·25-s − 0.177·26-s + 1.07·27-s − 0.598·28-s + 0.923·29-s + 0.318·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9745669862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9745669862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 13 | \( 1 + 0.907T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 + 6.25T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.172T + 43T^{2} \) |
| 47 | \( 1 + 5.76T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 0.804T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 3.19T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 4.04T + 97T^{2} \) |
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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
−8.333961657330939431840624794264, −7.17239904882473411838942814525, −6.83144230009078161975490420358, −6.08181972131693569280787679937, −5.36737258865385794298102507699, −4.72755162388648794998843686139, −3.66824678442263705384786806712, −3.20825899047188115711920375575, −2.12608336115462903984212899735, −0.48244884115906576873540741227,
0.48244884115906576873540741227, 2.12608336115462903984212899735, 3.20825899047188115711920375575, 3.66824678442263705384786806712, 4.72755162388648794998843686139, 5.36737258865385794298102507699, 6.08181972131693569280787679937, 6.83144230009078161975490420358, 7.17239904882473411838942814525, 8.333961657330939431840624794264
