| L(s) = 1 | + 2.79·2-s − 3-s + 5.80·4-s + 2.65·5-s − 2.79·6-s − 7-s + 10.6·8-s + 9-s + 7.41·10-s − 3.78·11-s − 5.80·12-s + 1.27·13-s − 2.79·14-s − 2.65·15-s + 18.0·16-s + 1.01·17-s + 2.79·18-s − 2.44·19-s + 15.4·20-s + 21-s − 10.5·22-s − 1.91·23-s − 10.6·24-s + 2.04·25-s + 3.57·26-s − 27-s − 5.80·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.90·4-s + 1.18·5-s − 1.14·6-s − 0.377·7-s + 3.75·8-s + 0.333·9-s + 2.34·10-s − 1.14·11-s − 1.67·12-s + 0.354·13-s − 0.746·14-s − 0.685·15-s + 4.52·16-s + 0.246·17-s + 0.658·18-s − 0.560·19-s + 3.44·20-s + 0.218·21-s − 2.25·22-s − 0.398·23-s − 2.17·24-s + 0.408·25-s + 0.700·26-s − 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.435165566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.435165566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 - 0.564T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 + 6.12T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 + 3.82T + 79T^{2} \) |
| 83 | \( 1 + 0.430T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−7.45370135526503384660753201026, −6.66146452905509783939515635646, −6.18041165543546331041015520179, −5.57889048081814032627023751551, −5.27753792132539834529148209137, −4.36704126447701000790664876350, −3.73352221207464628851982359551, −2.56098639103692201013039883193, −2.36183877317025663111340967510, −1.15699926036430818926420191957,
1.15699926036430818926420191957, 2.36183877317025663111340967510, 2.56098639103692201013039883193, 3.73352221207464628851982359551, 4.36704126447701000790664876350, 5.27753792132539834529148209137, 5.57889048081814032627023751551, 6.18041165543546331041015520179, 6.66146452905509783939515635646, 7.45370135526503384660753201026
