| L(s) = 1 | + 2-s + 2·4-s + 2·7-s + 5·8-s + 3·9-s − 4·11-s − 2·13-s + 2·14-s + 5·16-s + 3·17-s + 3·18-s + 6·19-s − 4·22-s + 8·23-s − 2·26-s + 4·28-s + 18·29-s − 20·31-s + 10·32-s + 3·34-s + 6·36-s + 11·37-s + 6·38-s + 9·41-s − 4·43-s − 8·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.755·7-s + 1.76·8-s + 9-s − 1.20·11-s − 0.554·13-s + 0.534·14-s + 5/4·16-s + 0.727·17-s + 0.707·18-s + 1.37·19-s − 0.852·22-s + 1.66·23-s − 0.392·26-s + 0.755·28-s + 3.34·29-s − 3.59·31-s + 1.76·32-s + 0.514·34-s + 36-s + 1.80·37-s + 0.973·38-s + 1.40·41-s − 0.609·43-s − 1.20·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.477972768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.477972768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - 11 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.33112038729168263640636751165, −10.10015082217828999944431246530, −9.395327023802856567894428135612, −9.336864264650541008897033432223, −8.212008649394670232277399050386, −8.134005279940897243835588009731, −7.62567501985610028733689108645, −7.30231305795038479880177906080, −6.94380142280896450913929201412, −6.66415431427463290965848929672, −5.63402476117309804912883727040, −5.48142445831349732251106790332, −4.93956128784287728518583243545, −4.63251522624793846765833196404, −4.20031771808600927924622335608, −3.43064357335669774246762756604, −2.73057961931775557151792166753, −2.53157389423715846901762624518, −1.36473062318667112130856009604, −1.26217481548301620448055469066,
1.26217481548301620448055469066, 1.36473062318667112130856009604, 2.53157389423715846901762624518, 2.73057961931775557151792166753, 3.43064357335669774246762756604, 4.20031771808600927924622335608, 4.63251522624793846765833196404, 4.93956128784287728518583243545, 5.48142445831349732251106790332, 5.63402476117309804912883727040, 6.66415431427463290965848929672, 6.94380142280896450913929201412, 7.30231305795038479880177906080, 7.62567501985610028733689108645, 8.134005279940897243835588009731, 8.212008649394670232277399050386, 9.336864264650541008897033432223, 9.395327023802856567894428135612, 10.10015082217828999944431246530, 10.33112038729168263640636751165
