L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 2.24·11-s + 12-s − 5.65·13-s + 16-s − 2.58·17-s + 18-s − 6.82·19-s + 2.24·22-s − 3.17·23-s + 24-s − 5.65·26-s + 27-s + 2.58·29-s − 10.2·31-s + 32-s + 2.24·33-s − 2.58·34-s + 36-s + 0.242·37-s − 6.82·38-s − 5.65·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.676·11-s + 0.288·12-s − 1.56·13-s + 0.250·16-s − 0.627·17-s + 0.235·18-s − 1.56·19-s + 0.478·22-s − 0.661·23-s + 0.204·24-s − 1.10·26-s + 0.192·27-s + 0.480·29-s − 1.83·31-s + 0.176·32-s + 0.390·33-s − 0.443·34-s + 0.166·36-s + 0.0398·37-s − 1.10·38-s − 0.905·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 9.07T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−7.50126380372864451004065470575, −6.80833715568616776066162151718, −6.23842567236345614791146785580, −5.35910037404707384005628595880, −4.44345868849569532157017353077, −4.15595320720763145372097341428, −3.13999479075993403204083811994, −2.31793932927312864603746475987, −1.73041078594936187790446366695, 0,
1.73041078594936187790446366695, 2.31793932927312864603746475987, 3.13999479075993403204083811994, 4.15595320720763145372097341428, 4.44345868849569532157017353077, 5.35910037404707384005628595880, 6.23842567236345614791146785580, 6.80833715568616776066162151718, 7.50126380372864451004065470575