| L(s) = 1 | + 1.89·3-s + 1.61·5-s − 3.11·7-s + 0.587·9-s − 1.98·11-s + 1.24·13-s + 3.06·15-s − 1.76·17-s − 0.148·19-s − 5.89·21-s + 7.88·23-s − 2.38·25-s − 4.56·27-s + 1.36·29-s − 11.0·31-s − 3.75·33-s − 5.03·35-s − 4.16·37-s + 2.34·39-s + 7.41·41-s − 2.06·43-s + 0.951·45-s − 47-s + 2.68·49-s − 3.34·51-s + 6.63·53-s − 3.20·55-s + ⋯ |
| L(s) = 1 | + 1.09·3-s + 0.723·5-s − 1.17·7-s + 0.195·9-s − 0.597·11-s + 0.343·13-s + 0.791·15-s − 0.427·17-s − 0.0340·19-s − 1.28·21-s + 1.64·23-s − 0.476·25-s − 0.879·27-s + 0.253·29-s − 1.98·31-s − 0.653·33-s − 0.851·35-s − 0.685·37-s + 0.376·39-s + 1.15·41-s − 0.314·43-s + 0.141·45-s − 0.145·47-s + 0.383·49-s − 0.467·51-s + 0.910·53-s − 0.432·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 - 1.89T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 0.148T + 19T^{2} \) |
| 23 | \( 1 - 7.88T + 23T^{2} \) |
| 29 | \( 1 - 1.36T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 4.16T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 8.76T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 6.57T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 3.33T + 89T^{2} \) |
| 97 | \( 1 - 5.90T + 97T^{2} \) |
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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
−7.60425302561634265477005501092, −7.22043440460196949228660937627, −6.22359288376239269178415358777, −5.74893998133125960195640219304, −4.82944855625820916548657607641, −3.72545323984928998001473421432, −3.11961606933572656388565099113, −2.51329146469810256374769081376, −1.59929898135947748720608606748, 0,
1.59929898135947748720608606748, 2.51329146469810256374769081376, 3.11961606933572656388565099113, 3.72545323984928998001473421432, 4.82944855625820916548657607641, 5.74893998133125960195640219304, 6.22359288376239269178415358777, 7.22043440460196949228660937627, 7.60425302561634265477005501092
