| L(s) = 1 | + 2·9-s + 12·17-s − 20·25-s + 22·49-s − 32·61-s − 44·73-s − 15·81-s + 72·101-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.91·17-s − 4·25-s + 22/7·49-s − 4.09·61-s − 5.14·73-s − 5/3·81-s + 7.16·101-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904735809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904735809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 127 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.672061874151792143968298734310, −8.197166136460573239315518007286, −7.74057098591661363622282987027, −7.70217812489275921921244879116, −7.61631595988494673010127141725, −7.41742151558505521450733070441, −7.08497936416952254733109559369, −6.94509425852524969688763172845, −6.18175554752335399584078816365, −5.93689062924762537169690252088, −5.84837517283389699269946374500, −5.74436869095877790237327727324, −5.70268222407413260784561238337, −4.92834989301185977865003404654, −4.74211133192560811781261001211, −4.27670906700932501796249552188, −4.14887493503137114582602114231, −3.90959713555463200898393385726, −3.26302997517606004399166295917, −3.14210296018590805661309531729, −3.00052263060552618418182761369, −2.04937680750422649026388617953, −1.85587376102068615898652805644, −1.48830481180566274900136557302, −0.65945811695591830263193753544,
0.65945811695591830263193753544, 1.48830481180566274900136557302, 1.85587376102068615898652805644, 2.04937680750422649026388617953, 3.00052263060552618418182761369, 3.14210296018590805661309531729, 3.26302997517606004399166295917, 3.90959713555463200898393385726, 4.14887493503137114582602114231, 4.27670906700932501796249552188, 4.74211133192560811781261001211, 4.92834989301185977865003404654, 5.70268222407413260784561238337, 5.74436869095877790237327727324, 5.84837517283389699269946374500, 5.93689062924762537169690252088, 6.18175554752335399584078816365, 6.94509425852524969688763172845, 7.08497936416952254733109559369, 7.41742151558505521450733070441, 7.61631595988494673010127141725, 7.70217812489275921921244879116, 7.74057098591661363622282987027, 8.197166136460573239315518007286, 8.672061874151792143968298734310
