| L(s) = 1 | − 2-s + 4-s − 2·5-s + 2·7-s − 8-s + 2·10-s − 4·11-s − 2·14-s + 16-s + 6·19-s − 2·20-s + 4·22-s − 4·23-s − 25-s + 2·28-s + 8·29-s + 2·31-s − 32-s − 4·35-s − 6·37-s − 6·38-s + 2·40-s + 6·41-s − 8·43-s − 4·44-s + 4·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.534·14-s + 1/4·16-s + 1.37·19-s − 0.447·20-s + 0.852·22-s − 0.834·23-s − 1/5·25-s + 0.377·28-s + 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.676·35-s − 0.986·37-s − 0.973·38-s + 0.316·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s + 0.589·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.269521242557942527546097632052, −7.67810526429361530303881958849, −7.26527327420570512771741413191, −6.12325887488210462294606111189, −5.22763893268397671843428475320, −4.49856970906518551878841710947, −3.39420084962357941083913884688, −2.52833574170498838498725263706, −1.30665134890636625066764125837, 0,
1.30665134890636625066764125837, 2.52833574170498838498725263706, 3.39420084962357941083913884688, 4.49856970906518551878841710947, 5.22763893268397671843428475320, 6.12325887488210462294606111189, 7.26527327420570512771741413191, 7.67810526429361530303881958849, 8.269521242557942527546097632052
