L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s + 3·7-s − 2.23·8-s + 9-s − 0.618·12-s + 6.23·13-s + 4.85·14-s − 4.85·16-s − 0.618·17-s + 1.61·18-s + 0.854·19-s − 3·21-s + 5.47·23-s + 2.23·24-s + 10.0·26-s − 27-s + 1.85·28-s + 4.47·29-s − 3.85·31-s − 3.38·32-s − 1.00·34-s + 0.618·36-s + 4.23·37-s + 1.38·38-s + ⋯ |
L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s + 1.13·7-s − 0.790·8-s + 0.333·9-s − 0.178·12-s + 1.72·13-s + 1.29·14-s − 1.21·16-s − 0.149·17-s + 0.381·18-s + 0.195·19-s − 0.654·21-s + 1.14·23-s + 0.456·24-s + 1.97·26-s − 0.192·27-s + 0.350·28-s + 0.830·29-s − 0.692·31-s − 0.597·32-s − 0.171·34-s + 0.103·36-s + 0.696·37-s + 0.224·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.644124182\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.644124182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 - 7.38T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.59164158949140146727177643022, −6.76397359468096519362730734716, −6.14954297646804587196055687028, −5.53314919091884724069737727790, −4.94980064952435933146497003433, −4.36766471185993990901504352530, −3.68263605532657912627169307790, −2.90373257030438294727383325499, −1.73075116385260903814366478978, −0.846717585528818081394688793010,
0.846717585528818081394688793010, 1.73075116385260903814366478978, 2.90373257030438294727383325499, 3.68263605532657912627169307790, 4.36766471185993990901504352530, 4.94980064952435933146497003433, 5.53314919091884724069737727790, 6.14954297646804587196055687028, 6.76397359468096519362730734716, 7.59164158949140146727177643022