| L(s) = 1 | + 0.454·2-s + 2.15·3-s − 1.79·4-s + 5-s + 0.977·6-s − 1.72·8-s + 1.63·9-s + 0.454·10-s − 3.04·11-s − 3.85·12-s + 1.17·13-s + 2.15·15-s + 2.80·16-s + 2.55·17-s + 0.740·18-s + 5.02·19-s − 1.79·20-s − 1.38·22-s − 0.685·23-s − 3.70·24-s + 25-s + 0.533·26-s − 2.94·27-s + 29-s + 0.977·30-s + 3.19·31-s + 4.72·32-s + ⋯ |
| L(s) = 1 | + 0.321·2-s + 1.24·3-s − 0.896·4-s + 0.447·5-s + 0.399·6-s − 0.609·8-s + 0.543·9-s + 0.143·10-s − 0.919·11-s − 1.11·12-s + 0.325·13-s + 0.555·15-s + 0.701·16-s + 0.620·17-s + 0.174·18-s + 1.15·19-s − 0.401·20-s − 0.295·22-s − 0.142·23-s − 0.756·24-s + 0.200·25-s + 0.104·26-s − 0.567·27-s + 0.185·29-s + 0.178·30-s + 0.574·31-s + 0.834·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.124625308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.124625308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.454T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 + 0.685T + 23T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 5.49T + 53T^{2} \) |
| 59 | \( 1 - 5.06T + 59T^{2} \) |
| 61 | \( 1 + 1.89T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 0.891T + 71T^{2} \) |
| 73 | \( 1 - 8.27T + 73T^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 0.210T + 97T^{2} \) |
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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
−8.062143382074931614089031994909, −7.56295804664686786636624333740, −6.47219241594175729966630398539, −5.61834975671054509940143218607, −5.11243105280834218389490802280, −4.27845853208286129243388390173, −3.32072191510154367661447430887, −3.02436608640464270562937059935, −2.02094991802965596965759247912, −0.813732222235736682471416673305,
0.813732222235736682471416673305, 2.02094991802965596965759247912, 3.02436608640464270562937059935, 3.32072191510154367661447430887, 4.27845853208286129243388390173, 5.11243105280834218389490802280, 5.61834975671054509940143218607, 6.47219241594175729966630398539, 7.56295804664686786636624333740, 8.062143382074931614089031994909
