| L(s) = 1 | + 12·17-s − 8·19-s − 10·25-s + 20·31-s + 2·49-s − 24·59-s + 20·61-s + 8·67-s − 24·71-s − 4·73-s − 20·79-s + 4·103-s − 24·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.91·17-s − 1.83·19-s − 2·25-s + 3.59·31-s + 2/7·49-s − 3.12·59-s + 2.56·61-s + 0.977·67-s − 2.84·71-s − 0.468·73-s − 2.25·79-s + 0.394·103-s − 2.32·107-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.204132151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.204132151\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.880406196379438251065378819208, −8.457415184628673389608811903699, −8.341840739673739455667916624142, −7.83097810479085325127171781506, −7.63843402015391526501072709137, −7.27029960141297032016128915688, −6.53955395913790917394545835636, −6.34896869458396410845913559818, −5.97645096262885397519981471185, −5.56587821216796220017967540132, −5.28238151014521328625690719694, −4.50119585284574641246200022451, −4.31941517583379993633591418436, −3.95720528457469954360782911870, −3.24018838777481419141260908090, −2.93870493198868333070785605224, −2.48747670729328495624732126227, −1.70054142344014590285747424452, −1.29353376480147041091862986209, −0.50757630156230118219999669991,
0.50757630156230118219999669991, 1.29353376480147041091862986209, 1.70054142344014590285747424452, 2.48747670729328495624732126227, 2.93870493198868333070785605224, 3.24018838777481419141260908090, 3.95720528457469954360782911870, 4.31941517583379993633591418436, 4.50119585284574641246200022451, 5.28238151014521328625690719694, 5.56587821216796220017967540132, 5.97645096262885397519981471185, 6.34896869458396410845913559818, 6.53955395913790917394545835636, 7.27029960141297032016128915688, 7.63843402015391526501072709137, 7.83097810479085325127171781506, 8.341840739673739455667916624142, 8.457415184628673389608811903699, 8.880406196379438251065378819208
