| L(s) = 1 | + 2.68·2-s − 3-s + 5.18·4-s − 2.68·6-s − 3.20·7-s + 8.53·8-s + 9-s − 2.67·11-s − 5.18·12-s + 5.20·13-s − 8.58·14-s + 12.5·16-s + 3.17·17-s + 2.68·18-s + 6.80·19-s + 3.20·21-s − 7.17·22-s + 3.15·23-s − 8.53·24-s + 13.9·26-s − 27-s − 16.6·28-s − 29-s − 5.16·31-s + 16.4·32-s + 2.67·33-s + 8.50·34-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.59·4-s − 1.09·6-s − 1.21·7-s + 3.01·8-s + 0.333·9-s − 0.807·11-s − 1.49·12-s + 1.44·13-s − 2.29·14-s + 3.12·16-s + 0.769·17-s + 0.631·18-s + 1.56·19-s + 0.699·21-s − 1.53·22-s + 0.658·23-s − 1.74·24-s + 2.73·26-s − 0.192·27-s − 3.13·28-s − 0.185·29-s − 0.927·31-s + 2.90·32-s + 0.466·33-s + 1.45·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.869210943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.869210943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 19 | \( 1 - 6.80T + 19T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 - 9.53T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 + 0.381T + 67T^{2} \) |
| 71 | \( 1 - 0.102T + 71T^{2} \) |
| 73 | \( 1 + 0.859T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−9.278025037979379988055049185460, −7.76442622039470579751938368867, −7.23072674704611594660583795549, −6.22886166300091863765736480986, −5.85754577784754918849470331723, −5.20984883060195712811188984465, −4.18566483579185391495534664277, −3.34253200294769053878899160768, −2.82606152489861165313810162461, −1.23035895081009857568835435409,
1.23035895081009857568835435409, 2.82606152489861165313810162461, 3.34253200294769053878899160768, 4.18566483579185391495534664277, 5.20984883060195712811188984465, 5.85754577784754918849470331723, 6.22886166300091863765736480986, 7.23072674704611594660583795549, 7.76442622039470579751938368867, 9.278025037979379988055049185460
