L(s) = 1 | − 2-s + 0.158·3-s + 4-s + 3.07·5-s − 0.158·6-s − 1.35·7-s − 8-s − 2.97·9-s − 3.07·10-s + 0.158·12-s + 0.526·13-s + 1.35·14-s + 0.488·15-s + 16-s − 7.23·17-s + 2.97·18-s + 19-s + 3.07·20-s − 0.214·21-s + 2.09·23-s − 0.158·24-s + 4.46·25-s − 0.526·26-s − 0.948·27-s − 1.35·28-s + 0.277·29-s − 0.488·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.0916·3-s + 0.5·4-s + 1.37·5-s − 0.0648·6-s − 0.511·7-s − 0.353·8-s − 0.991·9-s − 0.973·10-s + 0.0458·12-s + 0.146·13-s + 0.361·14-s + 0.126·15-s + 0.250·16-s − 1.75·17-s + 0.701·18-s + 0.229·19-s + 0.688·20-s − 0.0468·21-s + 0.436·23-s − 0.0324·24-s + 0.893·25-s − 0.103·26-s − 0.182·27-s − 0.255·28-s + 0.0515·29-s − 0.0891·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.158T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 13 | \( 1 - 0.526T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 23 | \( 1 - 2.09T + 23T^{2} \) |
| 29 | \( 1 - 0.277T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 - 8.91T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 - 1.96T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 + 1.02T + 67T^{2} \) |
| 71 | \( 1 + 3.71T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 + 7.53T + 79T^{2} \) |
| 83 | \( 1 + 6.49T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.210709373708600574632292528055, −7.14768363732364721155122665426, −6.45130762980135111522236063296, −5.97410453150298184665247302060, −5.26353323774928263631667093809, −4.16872069938312179979479454655, −2.85771938702019310486390732629, −2.45977213814147081954855718044, −1.41630660413439487458459463387, 0,
1.41630660413439487458459463387, 2.45977213814147081954855718044, 2.85771938702019310486390732629, 4.16872069938312179979479454655, 5.26353323774928263631667093809, 5.97410453150298184665247302060, 6.45130762980135111522236063296, 7.14768363732364721155122665426, 8.210709373708600574632292528055