| L(s) = 1 | − 1.04e3·7-s − 688·13-s − 4.64e3·19-s + 2.72e4·25-s + 2.11e4·31-s + 4.81e4·37-s − 1.81e5·43-s + 5.88e5·49-s − 5.02e5·61-s − 4.32e5·67-s − 6.16e5·73-s + 1.08e6·79-s + 7.21e5·91-s − 7.43e4·97-s − 2.92e6·103-s + 2.87e6·109-s + 2.79e6·121-s + 127-s + 131-s + 4.86e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.05·7-s − 0.313·13-s − 0.676·19-s + 1.74·25-s + 0.709·31-s + 0.950·37-s − 2.28·43-s + 5.00·49-s − 2.21·61-s − 1.43·67-s − 1.58·73-s + 2.19·79-s + 0.956·91-s − 0.0814·97-s − 2.67·103-s + 2.21·109-s + 1.57·121-s + 2.06·133-s − 1.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.08664011916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08664011916\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 1088 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 524 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2794034 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 344 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9376 p^{2} T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2320 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 262974530 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 653627360 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10564 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24082 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2345166880 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 90952 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4927927970 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5643898400 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82775997074 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 251138 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 216088 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 253293689090 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 308176 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 540124 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 215387844910 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 944049116864 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 37168 T + p^{6} T^{2} )^{2} \) |
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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
−10.21503475444558190708530263731, −9.640331887996294713951347125292, −9.227282849030912695823453915701, −8.665236268987532315821820592657, −8.505039694131558134458698000156, −7.55753728883391248111323558358, −7.27005531441380484231783399801, −6.58254158948280514691330919801, −6.48319384859896210725075122432, −6.15176433891682660740565559887, −5.58181394557847400315367555228, −4.74392314866871536944431509657, −4.49038605577193523331668425286, −3.57141848465705096954004214275, −3.33061086079824348267164159869, −2.83395564494553859431208854810, −2.49888002446713561882238085546, −1.50813583654701782601079964628, −0.77607859000474277988439676141, −0.07814800974230791379961315150,
0.07814800974230791379961315150, 0.77607859000474277988439676141, 1.50813583654701782601079964628, 2.49888002446713561882238085546, 2.83395564494553859431208854810, 3.33061086079824348267164159869, 3.57141848465705096954004214275, 4.49038605577193523331668425286, 4.74392314866871536944431509657, 5.58181394557847400315367555228, 6.15176433891682660740565559887, 6.48319384859896210725075122432, 6.58254158948280514691330919801, 7.27005531441380484231783399801, 7.55753728883391248111323558358, 8.505039694131558134458698000156, 8.665236268987532315821820592657, 9.227282849030912695823453915701, 9.640331887996294713951347125292, 10.21503475444558190708530263731
