L(s) = 1 | + 5·5-s − 30·7-s + 50·11-s − 88·13-s − 74·17-s − 140·19-s + 80·23-s + 25·25-s + 234·29-s − 150·35-s + 116·37-s + 72·41-s − 280·43-s + 120·47-s + 557·49-s + 498·53-s + 250·55-s − 870·59-s + 650·61-s − 440·65-s − 420·67-s + 1.02e3·71-s − 322·73-s − 1.50e3·77-s − 160·79-s − 980·83-s − 370·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.61·7-s + 1.37·11-s − 1.87·13-s − 1.05·17-s − 1.69·19-s + 0.725·23-s + 1/5·25-s + 1.49·29-s − 0.724·35-s + 0.515·37-s + 0.274·41-s − 0.993·43-s + 0.372·47-s + 1.62·49-s + 1.29·53-s + 0.612·55-s − 1.91·59-s + 1.36·61-s − 0.839·65-s − 0.765·67-s + 1.70·71-s − 0.516·73-s − 2.22·77-s − 0.227·79-s − 1.29·83-s − 0.472·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.269597898\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.269597898\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
good | 7 | \( 1 + 30 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 88 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 140 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 116 T + p^{3} T^{2} \) |
| 41 | \( 1 - 72 T + p^{3} T^{2} \) |
| 43 | \( 1 + 280 T + p^{3} T^{2} \) |
| 47 | \( 1 - 120 T + p^{3} T^{2} \) |
| 53 | \( 1 - 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 870 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 + 420 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 980 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1124 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1114 T + p^{3} T^{2} \) |
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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
−9.170770449175621346641164037392, −8.698067835935893108881640921367, −7.25145577137105647403766833281, −6.56085985836188356127498568580, −6.25861228117705552377472002338, −4.86388977716025109580838227857, −4.09493565939268740159186022999, −2.91591103789219758503069008077, −2.14227206263947767241181183518, −0.52728645845441789911851267592,
0.52728645845441789911851267592, 2.14227206263947767241181183518, 2.91591103789219758503069008077, 4.09493565939268740159186022999, 4.86388977716025109580838227857, 6.25861228117705552377472002338, 6.56085985836188356127498568580, 7.25145577137105647403766833281, 8.698067835935893108881640921367, 9.170770449175621346641164037392