L(s) = 1 | + (0.955 − 0.294i)4-s + (−0.988 + 0.149i)7-s + (−0.658 + 1.67i)13-s + (0.826 − 0.563i)16-s + 1.24·19-s + (0.365 + 0.930i)25-s + (−0.900 + 0.433i)28-s + (0.222 + 0.385i)31-s + (0.0990 + 0.433i)37-s + (1.03 − 0.702i)43-s + (0.955 − 0.294i)49-s + (−0.134 + 1.79i)52-s + (−0.425 − 0.131i)61-s + (0.623 − 0.781i)64-s + (0.900 + 1.56i)67-s + ⋯ |
L(s) = 1 | + (0.955 − 0.294i)4-s + (−0.988 + 0.149i)7-s + (−0.658 + 1.67i)13-s + (0.826 − 0.563i)16-s + 1.24·19-s + (0.365 + 0.930i)25-s + (−0.900 + 0.433i)28-s + (0.222 + 0.385i)31-s + (0.0990 + 0.433i)37-s + (1.03 − 0.702i)43-s + (0.955 − 0.294i)49-s + (−0.134 + 1.79i)52-s + (−0.425 − 0.131i)61-s + (0.623 − 0.781i)64-s + (0.900 + 1.56i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.455287834\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.455287834\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.988 - 0.149i)T \) |
good | 2 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 5 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (0.658 - 1.67i)T + (-0.733 - 0.680i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 31 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 43 | \( 1 + (-1.03 + 0.702i)T + (0.365 - 0.930i)T^{2} \) |
| 47 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 89 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.911001117834751917574931822008, −7.72389418853371288591877475082, −6.99465220896119875815332166785, −6.73129805563109034960185851339, −5.80584157529187782627980453004, −5.12330970725774536784798233628, −4.01859605792440987381374760047, −3.10175793396063602216324249696, −2.36374900731229685864484105466, −1.30885027836193699996835609299,
0.863164190853576811472320553346, 2.43102011012349232524326126668, 3.00328845263540959969064255759, 3.68312812915515800875959997910, 4.90917854971349468326665219709, 5.81856350244302655194761407476, 6.30556429736864692258154271541, 7.27801343983316549928418525477, 7.63212601854381098384103738705, 8.388467149206983180187350330038