| L(s) = 1 | + 3-s + 4.13·5-s + 9-s + 2.29·11-s − 2.43·13-s + 4.13·15-s + 3.71·17-s + 7.84·19-s + 1.70·23-s + 12.0·25-s + 27-s + 3.25·29-s − 1.80·31-s + 2.29·33-s − 5.65·37-s − 2.43·39-s − 6.81·41-s − 5.44·43-s + 4.13·45-s − 13.2·47-s + 3.71·51-s + 9.09·53-s + 9.50·55-s + 7.84·57-s − 14.9·59-s + 2.16·61-s − 10.0·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.84·5-s + 0.333·9-s + 0.693·11-s − 0.674·13-s + 1.06·15-s + 0.900·17-s + 1.80·19-s + 0.354·23-s + 2.41·25-s + 0.192·27-s + 0.603·29-s − 0.324·31-s + 0.400·33-s − 0.929·37-s − 0.389·39-s − 1.06·41-s − 0.829·43-s + 0.616·45-s − 1.93·47-s + 0.519·51-s + 1.24·53-s + 1.28·55-s + 1.03·57-s − 1.94·59-s + 0.277·61-s − 1.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.002289984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.002289984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.13T + 5T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 7.84T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 1.80T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 9.09T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 3.95T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.454354936382086483114900281001, −7.49269861957768744147744540121, −6.83482517945793104112583775173, −6.14797572370043070934174795386, −5.25031010699411657681864936736, −4.91451343226632032761867945045, −3.42993025958501169400738866604, −2.91263910094936057142436797505, −1.82890126691397605728281817187, −1.23360813198237427455226400396,
1.23360813198237427455226400396, 1.82890126691397605728281817187, 2.91263910094936057142436797505, 3.42993025958501169400738866604, 4.91451343226632032761867945045, 5.25031010699411657681864936736, 6.14797572370043070934174795386, 6.83482517945793104112583775173, 7.49269861957768744147744540121, 8.454354936382086483114900281001
