| L(s) = 1 | − 28·31-s − 52·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 5.02·31-s − 6.65·61-s − 81-s + 4·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124306659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124306659\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 337 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 2903 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 9743 T^{4} + p^{4} T^{8} \) |
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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
−7.08787575767419269011537394083, −6.69795574940877162864770533604, −6.65373616702408974849958648322, −6.25905167090188330134935932664, −5.89721627387279533303010670683, −5.85405537714350411410635522330, −5.76845981431194705053831199387, −5.45370120231366968070528596503, −5.22866315630624823397755240879, −4.92833393825861352073464548962, −4.69769972474846764067093226777, −4.36014247350366254408309390724, −4.33012882558652088259908790023, −3.86750781616940135731603473403, −3.81736524990983242471248363028, −3.33454330567655595976170662630, −3.11498564301799755441079090310, −2.95280983750729495426076496258, −2.87485216423345783022698885508, −2.05968572713021108732908885601, −1.77087649217755426898903470981, −1.71562304189961200071544134476, −1.66488608485280322720609565840, −0.72206556061217361207384685489, −0.26700128743525946645959404261,
0.26700128743525946645959404261, 0.72206556061217361207384685489, 1.66488608485280322720609565840, 1.71562304189961200071544134476, 1.77087649217755426898903470981, 2.05968572713021108732908885601, 2.87485216423345783022698885508, 2.95280983750729495426076496258, 3.11498564301799755441079090310, 3.33454330567655595976170662630, 3.81736524990983242471248363028, 3.86750781616940135731603473403, 4.33012882558652088259908790023, 4.36014247350366254408309390724, 4.69769972474846764067093226777, 4.92833393825861352073464548962, 5.22866315630624823397755240879, 5.45370120231366968070528596503, 5.76845981431194705053831199387, 5.85405537714350411410635522330, 5.89721627387279533303010670683, 6.25905167090188330134935932664, 6.65373616702408974849958648322, 6.69795574940877162864770533604, 7.08787575767419269011537394083
