L(s) = 1 | + 2.74·2-s + 0.0877·3-s + 5.52·4-s + 0.240·6-s + 1.38·7-s + 9.67·8-s − 2.99·9-s − 1.68·11-s + 0.484·12-s + 1.34·13-s + 3.80·14-s + 15.4·16-s + 3.06·17-s − 8.20·18-s + 8.07·19-s + 0.121·21-s − 4.62·22-s + 2.18·23-s + 0.848·24-s + 3.69·26-s − 0.525·27-s + 7.67·28-s − 8.72·29-s + 7.09·31-s + 23.1·32-s − 0.147·33-s + 8.41·34-s + ⋯ |
L(s) = 1 | + 1.94·2-s + 0.0506·3-s + 2.76·4-s + 0.0982·6-s + 0.524·7-s + 3.42·8-s − 0.997·9-s − 0.508·11-s + 0.139·12-s + 0.373·13-s + 1.01·14-s + 3.87·16-s + 0.743·17-s − 1.93·18-s + 1.85·19-s + 0.0265·21-s − 0.986·22-s + 0.455·23-s + 0.173·24-s + 0.723·26-s − 0.101·27-s + 1.45·28-s − 1.61·29-s + 1.27·31-s + 4.09·32-s − 0.0257·33-s + 1.44·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.362565572\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.362565572\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 - 0.0877T + 3T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 1.34T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 - 2.18T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 - 0.976T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 + 8.16T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 - 2.45T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 - 9.10T + 71T^{2} \) |
| 73 | \( 1 + 4.03T + 73T^{2} \) |
| 79 | \( 1 + 0.0889T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.520T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.73210815584656109537918992893, −7.29836865089081023396866278285, −6.28855346144780669301535298623, −5.71265919718386329043466807968, −5.15885819214785430906461699289, −4.66226782369405731330378580821, −3.37816592005364724335776840677, −3.27360207020829919769945207249, −2.25986582712691220687309921415, −1.24369350431582714728225722955,
1.24369350431582714728225722955, 2.25986582712691220687309921415, 3.27360207020829919769945207249, 3.37816592005364724335776840677, 4.66226782369405731330378580821, 5.15885819214785430906461699289, 5.71265919718386329043466807968, 6.28855346144780669301535298623, 7.29836865089081023396866278285, 7.73210815584656109537918992893