L(s) = 1 | + 5·2-s − 103·4-s − 930·7-s − 1.15e3·8-s − 8.45e3·11-s − 6.22e3·13-s − 4.65e3·14-s + 7.40e3·16-s + 9.59e3·17-s − 4.58e4·19-s − 4.22e4·22-s + 1.02e5·23-s − 3.11e4·26-s + 9.57e4·28-s − 8.75e4·29-s − 7.62e4·31-s + 1.84e5·32-s + 4.79e4·34-s − 2.64e5·37-s − 2.29e5·38-s − 1.03e5·41-s + 3.24e5·43-s + 8.70e5·44-s + 5.10e5·46-s − 8.55e5·47-s + 4.13e4·49-s + 6.40e5·52-s + ⋯ |
L(s) = 1 | + 0.441·2-s − 0.804·4-s − 1.02·7-s − 0.797·8-s − 1.91·11-s − 0.785·13-s − 0.452·14-s + 0.452·16-s + 0.473·17-s − 1.53·19-s − 0.845·22-s + 1.75·23-s − 0.347·26-s + 0.824·28-s − 0.666·29-s − 0.459·31-s + 0.997·32-s + 0.209·34-s − 0.858·37-s − 0.678·38-s − 0.234·41-s + 0.622·43-s + 1.54·44-s + 0.773·46-s − 1.20·47-s + 0.0502·49-s + 0.631·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.5570546454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5570546454\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 5 T + p^{7} T^{2} \) |
| 7 | \( 1 + 930 T + p^{7} T^{2} \) |
| 11 | \( 1 + 8450 T + p^{7} T^{2} \) |
| 13 | \( 1 + 6220 T + p^{7} T^{2} \) |
| 17 | \( 1 - 9590 T + p^{7} T^{2} \) |
| 19 | \( 1 + 45884 T + p^{7} T^{2} \) |
| 23 | \( 1 - 4440 p T + p^{7} T^{2} \) |
| 29 | \( 1 + 87550 T + p^{7} T^{2} \) |
| 31 | \( 1 + 76212 T + p^{7} T^{2} \) |
| 37 | \( 1 + 264440 T + p^{7} T^{2} \) |
| 41 | \( 1 + 103600 T + p^{7} T^{2} \) |
| 43 | \( 1 - 324680 T + p^{7} T^{2} \) |
| 47 | \( 1 + 855880 T + p^{7} T^{2} \) |
| 53 | \( 1 - 958190 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1239550 T + p^{7} T^{2} \) |
| 61 | \( 1 - 628522 T + p^{7} T^{2} \) |
| 67 | \( 1 + 310380 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3934300 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4556090 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5371644 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6711060 T + p^{7} T^{2} \) |
| 89 | \( 1 + 3346500 T + p^{7} T^{2} \) |
| 97 | \( 1 + 15829730 T + p^{7} T^{2} \) |
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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
−10.80971802689273392069512121898, −10.01757560425010983229840345567, −9.097022938426826692766196766145, −8.059269350984566446003762648317, −6.87836328233900367540045857610, −5.56462951670308995289545890746, −4.84749856739266761255091462792, −3.47624175441204271564785757125, −2.51772020449026865343551341137, −0.34160924155449154180017928825,
0.34160924155449154180017928825, 2.51772020449026865343551341137, 3.47624175441204271564785757125, 4.84749856739266761255091462792, 5.56462951670308995289545890746, 6.87836328233900367540045857610, 8.059269350984566446003762648317, 9.097022938426826692766196766145, 10.01757560425010983229840345567, 10.80971802689273392069512121898