L(s) = 1 | + 2.67·2-s + 0.481·3-s + 5.15·4-s + 1.28·6-s + 0.806·7-s + 8.44·8-s − 2.76·9-s + 3.67·11-s + 2.48·12-s + 2.15·14-s + 12.2·16-s + 1.35·17-s − 7.40·18-s + 1.67·19-s + 0.387·21-s + 9.83·22-s − 6.48·23-s + 4.06·24-s − 2.77·27-s + 4.15·28-s + 2.41·29-s + 5.28·31-s + 15.9·32-s + 1.76·33-s + 3.61·34-s − 14.2·36-s + 3.76·37-s + ⋯ |
L(s) = 1 | + 1.89·2-s + 0.277·3-s + 2.57·4-s + 0.525·6-s + 0.304·7-s + 2.98·8-s − 0.922·9-s + 1.10·11-s + 0.716·12-s + 0.576·14-s + 3.06·16-s + 0.327·17-s − 1.74·18-s + 0.384·19-s + 0.0846·21-s + 2.09·22-s − 1.35·23-s + 0.829·24-s − 0.534·27-s + 0.785·28-s + 0.449·29-s + 0.949·31-s + 2.81·32-s + 0.307·33-s + 0.619·34-s − 2.37·36-s + 0.619·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.946949997\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.946949997\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 - 0.481T + 3T^{2} \) |
| 7 | \( 1 - 0.806T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 - 3.19T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 - 3.84T + 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 + 1.87T + 97T^{2} \) |
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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
−8.056975373868678752029447145268, −7.56676104388186711205414479596, −6.50364665895862937249308610571, −6.07410100160330340323627178450, −5.43410547460155829752340443316, −4.46547358611955730183446072850, −3.98294017014005126431556774150, −3.08422430072582883988036119365, −2.43396968001122771374936446677, −1.35012550177475427936706341263,
1.35012550177475427936706341263, 2.43396968001122771374936446677, 3.08422430072582883988036119365, 3.98294017014005126431556774150, 4.46547358611955730183446072850, 5.43410547460155829752340443316, 6.07410100160330340323627178450, 6.50364665895862937249308610571, 7.56676104388186711205414479596, 8.056975373868678752029447145268