L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s − 11-s + 2·14-s − 16-s + 4·17-s + 22-s − 9·23-s + 2·28-s − 2·29-s − 8·31-s − 5·32-s − 4·34-s − 37-s − 6·41-s − 8·43-s + 44-s + 9·46-s − 8·47-s − 3·49-s − 2·53-s − 6·56-s + 2·58-s − 12·59-s − 7·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 0.301·11-s + 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.213·22-s − 1.87·23-s + 0.377·28-s − 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.685·34-s − 0.164·37-s − 0.937·41-s − 1.21·43-s + 0.150·44-s + 1.32·46-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.801·56-s + 0.262·58-s − 1.56·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.43807750848295, −14.83091029322032, −14.25139687678002, −13.78147840842677, −13.34180732296689, −12.69210823481422, −12.36772336129686, −11.69267502270880, −11.00586896632450, −10.43017018870031, −9.907289820433290, −9.647929298071238, −9.096912706251131, −8.351805211581016, −7.966954967885903, −7.501339405558770, −6.762126667559730, −6.151819136749237, −5.464044443891991, −4.985219135122047, −4.139035997155745, −3.577733063290072, −3.031413567363085, −1.882965022490744, −1.428881111192167, 0, 0,
1.428881111192167, 1.882965022490744, 3.031413567363085, 3.577733063290072, 4.139035997155745, 4.985219135122047, 5.464044443891991, 6.151819136749237, 6.762126667559730, 7.501339405558770, 7.966954967885903, 8.351805211581016, 9.096912706251131, 9.647929298071238, 9.907289820433290, 10.43017018870031, 11.00586896632450, 11.69267502270880, 12.36772336129686, 12.69210823481422, 13.34180732296689, 13.78147840842677, 14.25139687678002, 14.83091029322032, 15.43807750848295