| L(s) = 1 | − 0.207·2-s + 2.64·3-s − 1.95·4-s − 1.74·5-s − 0.548·6-s + 3.85·7-s + 0.821·8-s + 3.97·9-s + 0.361·10-s + 4.06·11-s − 5.16·12-s − 4.76·13-s − 0.800·14-s − 4.60·15-s + 3.74·16-s + 4.65·17-s − 0.825·18-s + 5.84·19-s + 3.41·20-s + 10.1·21-s − 0.843·22-s − 6.57·23-s + 2.16·24-s − 1.96·25-s + 0.988·26-s + 2.58·27-s − 7.54·28-s + ⋯ |
| L(s) = 1 | − 0.146·2-s + 1.52·3-s − 0.978·4-s − 0.779·5-s − 0.223·6-s + 1.45·7-s + 0.290·8-s + 1.32·9-s + 0.114·10-s + 1.22·11-s − 1.49·12-s − 1.32·13-s − 0.213·14-s − 1.18·15-s + 0.935·16-s + 1.12·17-s − 0.194·18-s + 1.34·19-s + 0.762·20-s + 2.22·21-s − 0.179·22-s − 1.37·23-s + 0.442·24-s − 0.392·25-s + 0.193·26-s + 0.497·27-s − 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.714521953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714521953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 - T \) |
| good | 2 | \( 1 + 0.207T + 2T^{2} \) |
| 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 + 6.68T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 3.47T + 89T^{2} \) |
| 97 | \( 1 + 7.67T + 97T^{2} \) |
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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
−11.90333862287820177390572857856, −10.16332980633797147664457567093, −9.475683118957834686734665094991, −8.539691966320712627284516432165, −7.933677702392514957030347366419, −7.38449173815573576232876026560, −5.19674640512189357710443137263, −4.23131243482190279639236095864, −3.36300139547081717541589080806, −1.58622331074270900913905014815,
1.58622331074270900913905014815, 3.36300139547081717541589080806, 4.23131243482190279639236095864, 5.19674640512189357710443137263, 7.38449173815573576232876026560, 7.933677702392514957030347366419, 8.539691966320712627284516432165, 9.475683118957834686734665094991, 10.16332980633797147664457567093, 11.90333862287820177390572857856
