| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4.61·7-s − 8-s + 9-s + 0.854·11-s + 12-s + 5.61·13-s − 4.61·14-s + 16-s − 5.70·17-s − 18-s − 7.09·19-s + 4.61·21-s − 0.854·22-s + 8.09·23-s − 24-s − 5.61·26-s + 27-s + 4.61·28-s − 7.70·29-s + 2.47·31-s − 32-s + 0.854·33-s + 5.70·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.74·7-s − 0.353·8-s + 0.333·9-s + 0.257·11-s + 0.288·12-s + 1.55·13-s − 1.23·14-s + 0.250·16-s − 1.38·17-s − 0.235·18-s − 1.62·19-s + 1.00·21-s − 0.182·22-s + 1.68·23-s − 0.204·24-s − 1.10·26-s + 0.192·27-s + 0.872·28-s − 1.43·29-s + 0.444·31-s − 0.176·32-s + 0.148·33-s + 0.978·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.685699116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685699116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 0.854T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 8.09T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 0.618T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 4.56T + 89T^{2} \) |
| 97 | \( 1 - 8.18T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.60117460001814790123815808961, −8.972527151304895505297455197288, −8.808798108697884325141960292324, −8.043164029822116798756954868928, −7.09696044061933819000435744988, −6.13507820152330042751533510841, −4.79444024909693425982103301951, −3.87869696057505651806647739525, −2.29213383794753257312806161143, −1.36708512983003916590438478009,
1.36708512983003916590438478009, 2.29213383794753257312806161143, 3.87869696057505651806647739525, 4.79444024909693425982103301951, 6.13507820152330042751533510841, 7.09696044061933819000435744988, 8.043164029822116798756954868928, 8.808798108697884325141960292324, 8.972527151304895505297455197288, 10.60117460001814790123815808961
