| L(s) = 1 | − 9·9-s + 32·11-s − 8·19-s − 164·29-s − 16·31-s − 492·41-s + 286·49-s + 1.18e3·59-s + 1.14e3·61-s + 1.53e3·71-s − 816·79-s + 81·81-s + 1.02e3·89-s − 288·99-s + 1.33e3·101-s − 4.15e3·109-s − 1.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.877·11-s − 0.0965·19-s − 1.05·29-s − 0.0926·31-s − 1.87·41-s + 0.833·49-s + 2.61·59-s + 2.40·61-s + 2.56·71-s − 1.16·79-s + 1/9·81-s + 1.21·89-s − 0.292·99-s + 1.31·101-s − 3.65·109-s − 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.468·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.301530793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301530793\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 82 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 80170 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 115562 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 195050 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 574 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 571942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 466670 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1116678 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 510 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1561150 T^{2} + p^{6} T^{4} \) |
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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
−10.39440214595026171084965836447, −10.08925464537923838124710758917, −9.481481918143408538412420897794, −9.323707853672958177011675162978, −8.519696114425871390472065603652, −8.501449951883615070174018474750, −7.929228793889385178004885778900, −7.26368244895566834683292449706, −6.75457906899322039530647368671, −6.67712189287014130888119718729, −5.87403071882933770742035925329, −5.36659441327925108958476994143, −5.11891543963230836061380993778, −4.27404490811162402601632327544, −3.68203004603728928782240578666, −3.54358853631334985851693434924, −2.48838017547789598607606052442, −2.07097017686324727908071753403, −1.21863089023482092184466507891, −0.47834145687975776989485504512,
0.47834145687975776989485504512, 1.21863089023482092184466507891, 2.07097017686324727908071753403, 2.48838017547789598607606052442, 3.54358853631334985851693434924, 3.68203004603728928782240578666, 4.27404490811162402601632327544, 5.11891543963230836061380993778, 5.36659441327925108958476994143, 5.87403071882933770742035925329, 6.67712189287014130888119718729, 6.75457906899322039530647368671, 7.26368244895566834683292449706, 7.929228793889385178004885778900, 8.501449951883615070174018474750, 8.519696114425871390472065603652, 9.323707853672958177011675162978, 9.481481918143408538412420897794, 10.08925464537923838124710758917, 10.39440214595026171084965836447
