| L(s) = 1 | − 2·4-s + 10·7-s + 10·9-s − 26·11-s + 4·16-s + 22·17-s + 58·19-s + 32·25-s − 20·28-s − 20·36-s + 94·37-s + 70·41-s + 52·44-s − 23·49-s + 22·53-s + 100·63-s − 8·64-s − 44·68-s − 86·71-s − 116·76-s − 260·77-s − 44·79-s + 19·81-s + 280·83-s − 248·89-s − 80·97-s − 260·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 10/7·7-s + 10/9·9-s − 2.36·11-s + 1/4·16-s + 1.29·17-s + 3.05·19-s + 1.27·25-s − 5/7·28-s − 5/9·36-s + 2.54·37-s + 1.70·41-s + 1.18·44-s − 0.469·49-s + 0.415·53-s + 1.58·63-s − 1/8·64-s − 0.647·68-s − 1.21·71-s − 1.52·76-s − 3.37·77-s − 0.556·79-s + 0.234·81-s + 3.37·83-s − 2.78·89-s − 0.824·97-s − 2.62·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376996 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.510160819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.510160819\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 307 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 320 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 608 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1904 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1520 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2080 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6512 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7154 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6800 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 43 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 5542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 140 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 124 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
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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
−10.43599040141691277377986410599, −10.32788451094803174807925857223, −9.636137708377242753389977837555, −9.607833558023579856122323549883, −8.971200760365194384192341178337, −8.174928916032482616348805838595, −7.85684710948024620294396327340, −7.67551895145357083034079719354, −7.46367152006068104918234385082, −6.79321178707220736336717026757, −5.68152437528885064018545581035, −5.60461027735090583218931712439, −5.14375235218905941893175709911, −4.53042765309234599173209286983, −4.45424963511720286564793311718, −3.18412241629072026123340477827, −3.06748873436378913161327226898, −2.19901762204570409839326621571, −1.12381162638045259243943202655, −0.910529985691767800832238684377,
0.910529985691767800832238684377, 1.12381162638045259243943202655, 2.19901762204570409839326621571, 3.06748873436378913161327226898, 3.18412241629072026123340477827, 4.45424963511720286564793311718, 4.53042765309234599173209286983, 5.14375235218905941893175709911, 5.60461027735090583218931712439, 5.68152437528885064018545581035, 6.79321178707220736336717026757, 7.46367152006068104918234385082, 7.67551895145357083034079719354, 7.85684710948024620294396327340, 8.174928916032482616348805838595, 8.971200760365194384192341178337, 9.607833558023579856122323549883, 9.636137708377242753389977837555, 10.32788451094803174807925857223, 10.43599040141691277377986410599
