| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s + 2·11-s + 6·12-s − 4·14-s − 4·15-s + 5·16-s − 4·17-s + 6·18-s + 12·19-s − 6·20-s − 4·21-s + 4·22-s + 2·23-s + 8·24-s + 3·25-s + 4·27-s − 6·28-s + 4·29-s − 8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s + 1.73·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.970·17-s + 1.41·18-s + 2.75·19-s − 1.34·20-s − 0.872·21-s + 0.852·22-s + 0.417·23-s + 1.63·24-s + 3/5·25-s + 0.769·27-s − 1.13·28-s + 0.742·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.89752061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.89752061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.196703858796739188560153817739, −8.147727388350889110626847068559, −7.39236057178524008778085831621, −7.27772328324847647783486650611, −6.92629739094547524832819084248, −6.91835192007875470793633336188, −6.07394353634456213969136144769, −5.99623684548832612387888536126, −5.40171512132082605836717659148, −4.91646029302149526259207680574, −4.68973462011025983463594410702, −4.28719398580617276511140402556, −3.67968132351779898138760259877, −3.62714699078363177068996538825, −3.17349135854875669713303499094, −2.88694349437830212469365459044, −2.47199874588289937695838118664, −1.86622207930276235169241405344, −1.21176451144004195300134450377, −0.70811602349851989247150333848,
0.70811602349851989247150333848, 1.21176451144004195300134450377, 1.86622207930276235169241405344, 2.47199874588289937695838118664, 2.88694349437830212469365459044, 3.17349135854875669713303499094, 3.62714699078363177068996538825, 3.67968132351779898138760259877, 4.28719398580617276511140402556, 4.68973462011025983463594410702, 4.91646029302149526259207680574, 5.40171512132082605836717659148, 5.99623684548832612387888536126, 6.07394353634456213969136144769, 6.91835192007875470793633336188, 6.92629739094547524832819084248, 7.27772328324847647783486650611, 7.39236057178524008778085831621, 8.147727388350889110626847068559, 8.196703858796739188560153817739
