| L(s) = 1 | + 5-s + 4·7-s − 13-s − 2·17-s + 25-s − 2·29-s − 4·31-s + 4·35-s − 6·37-s + 6·41-s − 4·43-s + 4·47-s + 9·49-s − 10·53-s + 2·61-s − 65-s − 8·67-s − 4·71-s − 6·73-s − 8·79-s + 8·83-s − 2·85-s + 6·89-s − 4·91-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.277·13-s − 0.485·17-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.37·53-s + 0.256·61-s − 0.124·65-s − 0.977·67-s − 0.474·71-s − 0.702·73-s − 0.900·79-s + 0.878·83-s − 0.216·85-s + 0.635·89-s − 0.419·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.00347801122181, −14.52931744525226, −14.19930178171217, −13.65163192065839, −13.05560004596430, −12.55214775232516, −11.87269546266901, −11.45046864745656, −10.90907751115954, −10.48683647252571, −9.885108335041901, −9.036269401401378, −8.894642215507267, −8.086517808990904, −7.620030829719291, −7.112201912023611, −6.371016767874346, −5.709346131819944, −5.162172710647932, −4.659659598412185, −4.078799631445245, −3.233256220607669, −2.379136501673358, −1.804766080560995, −1.219875691158323, 0,
1.219875691158323, 1.804766080560995, 2.379136501673358, 3.233256220607669, 4.078799631445245, 4.659659598412185, 5.162172710647932, 5.709346131819944, 6.371016767874346, 7.112201912023611, 7.620030829719291, 8.086517808990904, 8.894642215507267, 9.036269401401378, 9.885108335041901, 10.48683647252571, 10.90907751115954, 11.45046864745656, 11.87269546266901, 12.55214775232516, 13.05560004596430, 13.65163192065839, 14.19930178171217, 14.52931744525226, 15.00347801122181
