| L(s) = 1 | + 2·7-s − 3·9-s − 2·11-s + 13-s − 6·17-s − 6·19-s − 8·23-s + 2·29-s + 10·31-s + 6·37-s − 6·41-s − 4·43-s + 2·47-s − 3·49-s − 6·53-s − 10·59-s − 2·61-s − 6·63-s − 10·67-s + 10·71-s − 2·73-s − 4·77-s − 4·79-s + 9·81-s + 6·83-s − 6·89-s + 2·91-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 0.603·11-s + 0.277·13-s − 1.45·17-s − 1.37·19-s − 1.66·23-s + 0.371·29-s + 1.79·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.824·53-s − 1.30·59-s − 0.256·61-s − 0.755·63-s − 1.22·67-s + 1.18·71-s − 0.234·73-s − 0.455·77-s − 0.450·79-s + 81-s + 0.658·83-s − 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.084211881637533400233455521527, −8.210386578951025425376746968028, −8.055165253946902453872699794833, −6.54008227134812511033157194684, −6.06067530786903932474099613902, −4.86841374594031472590441997440, −4.23131852783550662973025725232, −2.81405254559865384229150945233, −1.92855269843114352805299531729, 0,
1.92855269843114352805299531729, 2.81405254559865384229150945233, 4.23131852783550662973025725232, 4.86841374594031472590441997440, 6.06067530786903932474099613902, 6.54008227134812511033157194684, 8.055165253946902453872699794833, 8.210386578951025425376746968028, 9.084211881637533400233455521527
