| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 4·10-s + 10·13-s − 4·16-s − 10·17-s − 4·20-s − 25-s + 20·26-s − 8·32-s − 20·34-s + 14·37-s + 20·41-s − 2·50-s + 20·52-s + 18·53-s − 20·61-s − 8·64-s − 20·65-s − 20·68-s − 10·73-s + 28·74-s + 8·80-s − 9·81-s + 40·82-s + 20·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s + 2.77·13-s − 16-s − 2.42·17-s − 0.894·20-s − 1/5·25-s + 3.92·26-s − 1.41·32-s − 3.42·34-s + 2.30·37-s + 3.12·41-s − 0.282·50-s + 2.77·52-s + 2.47·53-s − 2.56·61-s − 64-s − 2.48·65-s − 2.42·68-s − 1.17·73-s + 3.25·74-s + 0.894·80-s − 81-s + 4.41·82-s + 2.16·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.475544992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.475544992\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.67599034945331494902288757434, −9.627981487260499818513518465931, −9.330205966327137550518449663569, −8.828725030890345368320182968308, −8.562429496725344523077579243404, −8.241725848428945553568723512438, −7.38671161124681677231404147115, −7.38248269074306429111710550001, −6.45133865268278004635174474375, −6.33769607777875777369798015571, −5.83895897040164102096025854604, −5.67053290322739446642823771052, −4.62836210447680006054129101883, −4.33533192766915908418348351732, −3.98861786409896714761697870325, −3.85994512540815833098706395062, −2.86041093076571684453023707476, −2.65249025480493873713176071184, −1.70548983414259502956387981068, −0.69954960253167344242305936241,
0.69954960253167344242305936241, 1.70548983414259502956387981068, 2.65249025480493873713176071184, 2.86041093076571684453023707476, 3.85994512540815833098706395062, 3.98861786409896714761697870325, 4.33533192766915908418348351732, 4.62836210447680006054129101883, 5.67053290322739446642823771052, 5.83895897040164102096025854604, 6.33769607777875777369798015571, 6.45133865268278004635174474375, 7.38248269074306429111710550001, 7.38671161124681677231404147115, 8.241725848428945553568723512438, 8.562429496725344523077579243404, 8.828725030890345368320182968308, 9.330205966327137550518449663569, 9.627981487260499818513518465931, 10.67599034945331494902288757434
